;)Math class was always so frustrating for me. Why are there contradicting price diagrams for the same ETF? We already said at the beginning of this problem that ???u=y'?? 15.7 Change of Variables - Whitman College Change of Variables: Homogeneous Differential Equation #4 The most general local transformations are contact transformations $x=X(x,y,y')$, $y=Y(x,y,y')$, $y'=P(x,y,y')$, with some conditions on the functions $X$, $Y$, $P$ to make the transformation make sense. Proceeding in this way it is possible to transform the second equation in the following way Differential Equations - Definition, Formula, Types, Examples - Cuemath First we need change the variable of differential equation . Try the free Mathway calculator and problem solver below to practice various math topics. A first attempt is to use a generic change of variables to identify the function F such that a ( ) = F ( y ( )). Connect and share knowledge within a single location that is structured and easy to search. DLMF: 1.13 Differential Equations Areas Chapter 1 Algebraic and =f(\zeta), with ???u???. You da real mvps! Stack Overflow for Teams is moving to its own domain! ?, we want to solve it for ???u'???. -y''(\zeta)\frac{F'(y(\zeta))}{2 F(y(\zeta))}+y'(\zeta)^2 \left(\frac{F'(y(\zeta))^2}{4 F(y(\zeta))^2}-\frac{F''(y(\zeta))}{2 F(y(\zeta))}\right)+\frac{c}{F(y(\zeta))^2}=f(\zeta). Since $F$ is presumed to be only a function of $y$ they can't be equal. Differential Equations - Introduction Read more. It is a variable that stands alone and isn't changed by the other variables you are trying to measure. Differential equation change of variables with sympy \end{equation}, \begin{equation} Since ???u=Q(x)-P(x)y?? Where to find hikes accessible in November and reachable by public transport from Denver? \end{equation} Differential to Difference equation with two variables? However, I really appreciated your suggestion to have a look to the Olver's book. 4) Movement of electricity can also be described with the help of it. Our aim is to find the general solution for the given differential equation . rev2022.11.7.43014. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company, \begin{equation} MIT, Apache, GNU, etc.) so that the equation becomes ???u=Q(x)-P(x)y???. Often, a first-order ODE that is neither separable nor linear can be simplified to one of these types by making a change of variables. Differential equation change of variables - Mathematics Stack Exchange \end{equation}. \end{equation} $1 per month helps!! dy/dx is not a quotient. I have an ordinary differential equation like this: DiffEq = Eq (-**diff (,x,2)/ (2*m) + m*w*w* (x*x)*/2 - E* , 0) I want to perform a variable change : sp.Eq (u , x*sqrt (m*w/)) sp.Eq (, H*exp (-u*u/2)) How can I do this with sympy? -y(\zeta) \left(\frac{d^2 y(\zeta)^{-1}}{d \zeta^2}+2 j c \frac{d y(\zeta)^{-1}}{d \zeta}\right)+c^2 (1+y(\zeta)^{2})=\\ Differential equation change of variables. Change variables in differential expressions - Mathematica Stack Exchange $$\frac{d^2y}{dx^2}=\frac{d}{dx}\bigg(\frac{dy}{dr}\bigg)\frac{dr}{dx}+\frac{dy}{dr}\frac{d}{dx}\bigg(\frac{dr}{dx}\bigg)$$ Change of Variables / Homo. \begin{equation} By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. Change of variables (PDE) Often a partial differential equation can be reduced to a simpler form with a known solution by a suitable change of variables . Use the indicated change of variables to find the general solution of \frac{y''(\zeta)}{y(\zeta)}-\frac{2 y'(\zeta)^2}{y(\zeta)^2}+\frac{2 c j y'(\zeta)}{y(\zeta)}+c^2 y(\zeta)^2+c^2=\\ -\frac{1}{2 a(\zeta)}\left(\frac{d^2 a(\zeta)}{d \zeta^2}-\frac{1}{2 a(\zeta)}\left(\frac{d a(\zeta)}{d \zeta}\right)^2 \right)+\frac{c}{a(\zeta)^2}=\\ Change of variables in partial derivatives - Online Technical To subscribe to this RSS feed, copy and paste this URL into your RSS reader. Given. in terms of ???x???. Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Try the given examples, or type in your own problem and check your answer with the step-by-step . In this case, it can be really helpful to use a change of variable to find the . Sometimes we'll be given a differential equation in the form???y'=Q(x)-P(x)y??? In this video, I solve a homogeneous differential equation by using a change of variables. Change of variable for differential equations - MathOverflow ?, so if we replace ???y'??? At this point we are two-thirds done with the task: we know the r - limits of integration, and we can easily convert the function to the new variables: x2 + y2 = r2cos2 + r2sin2 = rcos2 + sin2 = r. The final, and most difficult, task is to figure out what replaces dxdy. My profession is written "Unemployed" on my passport. Take the derivative of both sides in order to get ???y'???. First-Order Differential Equations - Calculus Tutorials =f(\zeta), Unfortunately, the transformation that would do the job is highly implicit in the proof and you shouldn't expect to be able to easily find its explicit form. Oct 5, 2017 - Change of Variables / Homogeneous Differential Equation - Example 1. Does subclassing int to forbid negative integers break Liskov Substitution Principle? Change of Variables in differential equation, Solution of differential equation- Change of variables, Change of Variables in a Second Order Linear Homogeneous Differential Equation, Variable Change In A Differential Equation, Variable change to make differential equation separable, Change of variables in a differential equation, Particular Reason for this Change of Variables in Ordinary Differential Equation. Since ???u'??? ?s on the right, well integrate both sides. JavaScript is disabled. Does the dependent variable change? Changing variables in separable DEs - Krista King Math Second Order Differential Equation - Change of Dependent Variable Method. python sympy differential-equations Share Improve this question edited Sep 8, 2019 at 12:07 Upax Asks: Change of variable for differential equations This question was previously posted here Change of variable for differential equations. on the left side with ???u?? -y(\zeta) \left(\frac{d^2 y(\zeta)^{-1}}{d \zeta^2}+2 j c \frac{d y(\zeta)^{-1}}{d \zeta}\right)+c^2 (1+y(\zeta)^{2})=\\ differential equations - How to Make a change of variables Proceeding in this way it is possible to transform the second equation DEFINITION 1.8.8 A differential equation that can be written in the form dy dx +p(x)y= q(x)yn, (1.8.9) where n is a real constant, is called a Bernoulli equation. For instance for $F=F(\zeta,y)$, I have Stack Overflow for Teams is moving to its own domain! Use the change of variables z = y x to convert the ODE to xdz dx = f(1, z) z, which is separable. Moreover, if the point transformation invariants of the OP's two equation can be shown to be different, it will also give a reason to stop looking for a point equivalence between them. Change of Variables / Homogeneous Differential Equation - YouTube Integrate both sides with respect to ???x?? Sometimes well be given a differential equation in the form. Homogeneous Differential Equations - Change of Variables . This book is a classic reference on the so called Cartan approach to equivalence problems, where the case of 2nd order ODEs is treated as an example: Olver, Peter J., Equivalence, invariants, and symmetry, Cambridge: Cambridge University Press (ISBN 978-0-521-10104-2/pbk). You sure that last term is $ay'(x)$ and not just $ay(x)$? ?, then the rest of the problem should fall into place. Change of variables is an operation that is related to substitution. \frac{d^{2}}{d \zeta^{2}} \log{y(\zeta)}-\left(\frac{d}{d \zeta} \log{y(\zeta)}\right)^{2}+2jc \frac{d}{d \zeta} \log{y(\zeta)}+c^2 y(\zeta)^2+c^2 !So I started tutoring to keep other people out of the same aggravating, time-sucking cycle. Change of variable for differential equations | SolveForum As for the sub I'd recommend to find $y_x$ through $y_r = y_x x_r$, not if that matters, just might cause less confusion. Change of variable to solve a differential equations - YouTube Here are some important examples: Homogeneous Equation of Order 0: dy dx = f(x, y) where f(kx, ky) = f(x, y). We can remove the absolute value brackets by adding a ???\pm??? What led you to believe such a change of variable exists? :) https://www.patreon.com/patrickjmt !! apply to documents without the need to be rewritten? I make math courses to keep you from banging your head against the wall. However, as of this point, I have no idea how to proceed. is a constant, so we can just call it ???C???. When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. Change of variables in a differential equation | Physics Forums ?, which is what we want, so well go ahead and substitute ???u??? @LSpice Sure, but I doubt very much that there is any explicit solution (that goes for the solutions of the original equation, as well as for a trivializing contact transformation). \frac{d^{2}}{d \zeta^{2}} \log{y(\zeta)}-\left(\frac{d}{d \zeta} \log{y(\zeta)}\right)^{2}+2jc \frac{d}{d \zeta} \log{y(\zeta)}+c^2 y(\zeta)^2+c^2 @MichaelAngelo By applying $\frac{d}{dx}=(\frac{dr}{dx})\frac{d}{dr}$ to both sides of the chain rule equation above, and using the product rule on the right hand side. rev2022.11.7.43014. Now we need to find the derivative of ???y?? In this video, I solve a homogeneous differential equation by using a change of variables. Remember, since ???u??? @Igor. in the following way ?, ???y'??? -y''(\zeta)\frac{F'(y(\zeta))}{2 F(y(\zeta))}+y'(\zeta)^2 \left(\frac{F'(y(\zeta))^2}{4 F(y(\zeta))^2}-\frac{F''(y(\zeta))}{2 F(y(\zeta))}\right)+\frac{c}{F(y(\zeta))^2}=f(\zeta). Also, I am not sure that changing variables $a(\zeta)=F(y(\zeta))$ is enough to solve the problem. Examples of separable differential equations include. Any suggestion is welcome. . Making statements based on opinion; back them up with references or personal experience. Differential equations Variable changes for differentiation and integration are taught in elementary calculus and the steps are rarely carried out in full. Well, I have tried it hard but I don't get the right result. I'm really interested in solving this problem, so if anything is unclear, please don't hesitate to let me know so that I can improve the post. Anyway, you have a good start. ?, we can make a substitution for ???u???. Learn more here: http://www.kristakingmath.comFACEBOOK // https://www.facebook.com/KristaKingMathTWITTER // https://twitter.com/KristaKingMathINSTAGRAM // https://www.instagram.com/kristakingmath/PINTEREST // https://www.pinterest.com/KristaKingMath/GOOGLE+ // https://plus.google.com/+Integralcalc/QUORA // https://www.quora.com/profile/Krista-King change of variables, differential equations, elimination of first derivative See also: Annotations for 1.13(iv), 1.13 and Ch.1. the following equation: You sure that last term is and not just ? Step 2: Assuming the form of the general solution. 2jc \frac{d}{d \zeta} \log{y(\zeta)}+c^2 y(\zeta)^2+c^2 =c_{1} y(\zeta)^4 for ???y'???. My Differential Equations course: https://www.kristakingmath.com/differential-equations-courseLearn how to use a change of variable to solve a separable di. Change of Variables / Homogeneous Differential Equation - Example 2. Free ebook https://bookboon.com/en/partial-differential-equations-ebook An example showing how to solve PDE via change of variables. Upax Asks: Change of variable for differential equations Given the following differential equation \begin{equation} -y(\zeta) \left(\frac{d^2. to find the general solution to the differential equation. However these are different operations, as can be seen when considering differentiation ( chain rule) or integration ( integration by substitution ). Is there a keyboard shortcut to save edited layers from the digitize toolbar in QGIS? Share Improve this answer Follow edited Apr 13, 2017 at 12:56 Community Bot 1 answered Mar 28, 2014 at 17:52 and asked to find a general solution to the equation, which will be an equation for ???y??? I doubt that a solution exists. The derivative represents nothing but a rate of change, and the differential equation helps us present a relationship between the changing quantity with respect to the change in another quantity. @fawningflagellum Please check the added section. Will it have a bad influence on getting a student visa? and its derivative ???u'???. It may not display this or other websites correctly. Sci-Fi Book With Cover Of A Person Driving A Ship Saying "Look Ma, No Hands!". It is a differential quotient. Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. ?, then replace ???u'??? Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site, Learn more about Stack Overflow the company. is the same thing as ???du/dx?? The main mathematical result in this paper is that change of variables in the ordinary differential equation (ODE) for the competition of two infections in a Susceptible-Infected-Removed (SIR) model shows that the fraction of cases due to the new variant satisfies the logistic differential equation, which models selective sweeps. in terms of ???x???. with ???du/dx???. If the general solution of this problem is difficult to determine, if it exists at all, it is probably possible to determine a particular solution. xvi, 525p. (2009). Id think, WHY didnt my teacher just tell me this in the first place? What is rate of emission of heat from a body in space? So you have a change of variables that looks like: x'=x' (x,y,t) y'=y' (x,y,t) t'=t Chain rule: df/dy = df/dx' * dx'/dy + df/dy'*dy'/dy + df/dt'*dt'/dy= sin (wt)df/dx' +cos (wt)df/dy' Sorry, I'm not sure how to use latex here. Asking for help, clarification, or responding to other answers. f(\zeta). The initial problem then reduces to identifying the function $F$ such that the first and the third equations are equal. The basic classic result is that every 2nd order ODE of the form $y'' = Q(x,y,y')$ is equivalent by a contact transformation to $y''=0$ [Olver, Thm.11.11]. 2jc \frac{d}{d \zeta} \log{y(\zeta)}+c^2 y(\zeta)^2+c^2 =c_{1} y(\zeta)^4 If this change of variables exists, how is it possible to find it? We need to change the current equation so that it is in terms of a new variable ???u??? $$x=r^t\Rightarrow dx=tr^{t-1}dr\Rightarrow\frac{dr}{dx}=\frac 1{tr^{t-1}}$$ u(x, t) may now be found simply by adding h(x), according to how the variable change was defined: u(x,t)=v(x,t)+h(x){\displaystyle u(x,t)=v(x,t)+h(x)\,} In this case, it can be really helpful to use a change of variable to find the solution. Use a change of variable to solve the differential equation. Which is the dependent variable? Now that the variables are separated, with the ???u?? q^2 -> 1 - usin^2 // Simplify The output is your desired result but in expanded form. \end{equation}, \begin{equation} Thanks for contributing an answer to Mathematics Stack Exchange! ?, we can change the equation to, Once you change variables and get the variables separated in the differential equation, then you can integrate both sides to find a soltuion. To learn more, see our tips on writing great answers. Change of variables (PDE) - Wikipedia I am trying witout success to make a change of variables in a partial derivative of a function of 2 variables (for example the time coordinate "t" and the lenght coordinate "z"), like. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. The equivalence problem is set up within the framework of Cartan's equivalence method in [Olver, Ex.9.3,9.6]. Site design / logo 2022 Stack Exchange Inc; user contributions licensed under CC BY-SA. Interested in getting help? Solve for ???y??? -\frac{y''(\zeta) F^{(0,1)}(\zeta,y(\zeta))}{2 F(\zeta,y(\zeta))}+y'(\zeta)^2 \left(\frac{F^{(0,1)}(\zeta,y(\zeta))^2}{4 F(\zeta,y(\zeta))^2}-\frac{F^{(0,2)}(\zeta,y(\zeta))}{2 F(\zeta,y(\zeta))}\right)+y'(\zeta) \left(\frac{F^{(0,1)}(\zeta,y(\zeta)) F^{(1,0)}(\zeta,y(\zeta))}{2 F(\zeta,y(\zeta))^2}-\frac{F^{(1,1)}(\zeta,y(\zeta))}{F(\zeta,y(\zeta))}\right)+\frac{F^{(1,0)}(\zeta,y(\zeta))^2}{4 F(\zeta,y(\zeta))^2}-\frac{F^{(2,0)}(\zeta ,y(\zeta))}{2 F(\zeta,y(\zeta))}+\frac{c}{F(\zeta,y(\zeta))^2}=f(\zeta). Prove this change with the following exercise: MathOverflow is a question and answer site for professional mathematicians. Differential Equations (Definition, Types, Order, Degree, Examples) - BYJUS For example, someone's age might be an independent variable. Consequences resulting from Yitang Zhang's latest claimed results on Landau-Siegel zeros. By clicking Accept all cookies, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy. The x 2 in b ( x) x 2 is nothing but the factor from coordinate transformation, wich makes b ( x) x 2 = d b ( 1 / x) / d x = [ d b ( r) / d r] / [ d r / d x] (where r = 1 / x ). Consider the identity relation d f ( r) = f ( r) d r = f ( r ( x)) r ( x) d x ==> f ( r) = f ( r ( x)). where $c$ is a constant, while $j=\sqrt{-1}$, Stack Exchange network consists of 182 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. We can include the ???\pm??? $$\frac{d^2y}{dx^2}=\frac{d^2y}{dr^2}\bigg(\frac{dr}{dx}\bigg)^2+\frac{dy}{dr}\frac{d^2r}{dx^2}$$ 2) They are also used to describe the change in return on investment over time. is there a change of variables that allows it to be transformed into the following form? 8 9. -\frac{d^{2}}{d \zeta^{2}} \log{\sqrt{a(\zeta)}}-\left(\frac{d}{d \zeta} \log{\sqrt{a(\zeta)}}\right)^{2}+\frac{c_{1}}{a(\zeta )^2}=\\ \frac{y''(\zeta)}{y(\zeta)}-\frac{2 y'(\zeta)^2}{y(\zeta)^2}+\frac{2 c j y'(\zeta)}{y(\zeta)}+c^2 y(\zeta)^2+c^2=\\ \begin{equation} \begin{equation} Change of Variables - Department of Mathematics at UTSA MathJax reference. Differential equation change of variables, Mobile app infrastructure being decommissioned. The second order differential equation y'' = f (t,y') y = f (t,y) can be solved making the change of variable z = y' \implies z' = y'' z = y z = y and, later, if we get a solution for z z, it will be sufficient to integrate \int z (t) \space dt z(t) dt to solve the initial equation. How can my Beastmaster ranger use its animal companion as a mount? I have the following differential equation, which is the general Sturm-Liouville problem, $$ and ???u'?? We now consider a special type of nonlinear differential equation that can be reduced to a linear equation by a change of variables. Finally, we solve for ???y??? By plugging into original equation Any help is welcome. \begin{equation} Since then, Ive recorded tons of videos and written out cheat-sheet style notes and formula sheets to help every math studentfrom basic middle school classes to advanced college calculusfigure out whats going on, understand the important concepts, and pass their classes, once and for all. The equivalence of (in general) non-linear 2nd order ordinary differential equations $y'' = Q(x,y,y')$ under various types of transformations (including the ones you are considering) is a classic problem. Is this claimed in a paper? This question was previously posted on MSE at Change of variable for differential equations. The steps for changing variables in a separable differential equation. 3) They are used in the field of medical science for modelling cancer growth or the spread of disease in the body. If you're looking for very explicit formulas for the invariants that can help you distinguish your two equations, then you might want to follow some of the references that Olver gives in that section. - Silvia Aug 13, 2012 at 18:07 Add a comment 0 Example of how to solve PDE via change of variables - YouTube In this video, I solve a homogeneous differential equation by using a change of variables. A first attempt is to use a generic change of variables to identify the function $F$ such that Since ???u=y'?? Calculus III - Change of Variables - Lamar University You are using an out of date browser. ?s on the left and the ???x?? ?, back-substitute and replace ???u??? When the migration is complete, you will access your Teams at stackoverflowteams.com, and they will no longer appear in the left sidebar on stackoverflow.com. Standard topology is coarser than lower limit topology? how does $\frac{d^2y}{dx^2}= \frac{d^2y}{dr^2}\bigg(\frac{dr}{dx}\bigg)^2+\frac{dy}{dr}\frac{d^2r}{dx^2}$ follow from the chain rule? My Differential Equations course: https://www.kristakingmath.com/differential-equations-courseLearn how to use a change of variable to solve a separable differential equation. GET EXTRA HELP If you could use some extra help with your math class, then check out Kristas website // http://www.kristakingmath.com CONNECT WITH KRISTA Hi, Im Krista! u(x,t) = (x)G(t) (1) (1) u ( x, t) = ( x) G ( t) will be a solution to a linear homogeneous partial differential equation in x x and t t. This is called a product solution and provided the boundary conditions are also linear and homogeneous this will also satisfy the boundary conditions. As pointed out by LSpice the question is about solving the equation. Why was video, audio and picture compression the poorest when storage space was the costliest? Is there any alternative way to eliminate CO2 buildup than by breathing or even an alternative to cellular respiration that don't produce CO2? with ???Q(x)-P(x)y???. on the other. .. totally wrong and this was a disaster. 8.3: Separable Differential Equations - Mathematics LibreTexts Finding the differential equation of motion. -\frac{y''(\zeta) F^{(0,1)}(\zeta,y(\zeta))}{2 F(\zeta,y(\zeta))}+y'(\zeta)^2 \left(\frac{F^{(0,1)}(\zeta,y(\zeta))^2}{4 F(\zeta,y(\zeta))^2}-\frac{F^{(0,2)}(\zeta,y(\zeta))}{2 F(\zeta,y(\zeta))}\right)+y'(\zeta) \left(\frac{F^{(0,1)}(\zeta,y(\zeta)) F^{(1,0)}(\zeta,y(\zeta))}{2 F(\zeta,y(\zeta))^2}-\frac{F^{(1,1)}(\zeta,y(\zeta))}{F(\zeta,y(\zeta))}\right)+\frac{F^{(1,0)}(\zeta,y(\zeta))^2}{4 F(\zeta,y(\zeta))^2}-\frac{F^{(2,0)}(\zeta ,y(\zeta))}{2 F(\zeta,y(\zeta))}+\frac{c}{F(\zeta,y(\zeta))^2}=f(\zeta). How can you prove that a certain file was downloaded from a certain website? Thanks to all of you who support me on Patreon. ZBL1156.58002. legal basis for "discretionary spending" vs. "mandatory spending" in the USA. I have a differential equation $$xy''(x) +(n+1-x)y'(x) + ay(x)=0.$$ The article discusses change of variable for PDEs below in two ways: by example; by giving the theory of the method. 2022 Physics Forums, All Rights Reserved, Change of variables in multiple integrals, Solving the wave equation with change of variables approach, Using separation of variables in solving partial differential equations. and asked to find a general solution to the equation, which will be an equation for ???y??? is a function, and not just a variable, its derivative is ???u'?? and ???u'?? Integral-form change of variable in differential equation I; Thread starter Jaime_mc2; Start date Jan 12, 2022; Tags change of variables differential equations Jan 12, 2022 #1 Jaime_mc2. Variable changes in linear differential equations of first order: y'=f Did the dependent variable change? - masx.afphila.com Advanced Mathematics for Engineers and Scientists/Change of Variables Change of variable for Jacobian: is there a method? fu:= f [t,z] dfu:= D [fu, { {t,z}}] Then I want to rescale the t and z coordinates (something that is useful for example to simplify equations in fluid mechanics . -\frac{d^{2}}{d \zeta^{2}} \log{\sqrt{a(\zeta)}}-\left(\frac{d}{d \zeta} \log{\sqrt{a(\zeta)}}\right)^{2}+\frac{c_{1}}{a(\zeta )^2}=\\ Now that our equation is entirely in terms of ???u??? Solving this equation for ???y?? A Differential Equation is a n equation with a function and one or more of its derivatives: . $$r^t \Bigg(\frac{d^2y}{dr^2}\bigg(\frac 1{tr^{t-1}}\bigg)^2+\frac{dy}{dr}\frac 1{t(t-1)r^{t-2}} \Bigg)+(n+1-r^t)\frac{dy}{dr}\frac 1{tr^{t-1}} + ay=0$$, ---- Addition for chain rule ---- Separate variables to put ???u??? These steps can be hard to remember and tricky to follow, but the key is to get rid of all of the ???y?? Did Great Valley Products demonstrate full motion video on an Amiga streaming from a SCSI hard disk in 1990? I have a differential equation If I set then how to plug in this and how to use change of variable to get the differential equation for instead of i.e. Integral-form change of variable in differential equation Why bad motor mounts cause the car to shake and vibrate at idle but not when you give it gas and increase the rpms? ?, back-substitute and replace ???y'??? Given the following differential equation (Nonetheless, a reference to Olver is always welcome.). Change of variable for differential equations | SolveForum
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