# Second-order Ordinary Differential Equations cheatsheet Star

## General case

General form The general form of a second-order ODE can be written as a function $F$ of $x, y, y'$ and $y''$ as follows:

$\boxed{F(x,y,y',y'')=0}$

Methods of resolution The table below summarizes the general tricks to apply when the ODE has the following classic forms:

 Old form Trick New form $\displaystyle F\left(x,y',y''\right)=0$ $\displaystyle y'\triangleq u$, $\quad\displaystyle y''=\frac{du}{dx}$ $G\left(x,u,\frac{du}{dx}\right)=0$ $\displaystyle F\left(y,y',y''\right)=0$ $\displaystyle y'\triangleq u$, $\quad\displaystyle y''=u\frac{du}{dy}$ $G\left(y,u,\frac{du}{dy}\right)=0$ $\displaystyle F\left(y',y''\right)=0$ $\displaystyle y'\triangleq u, \quad y''=\frac{du}{dx}$$\displaystyle y'\triangleq u,\quad y''=u\frac{du}{dy} Missing-y approach G\left(u,\frac{du}{dx}\right)=0Missing-x approach G\left(u,\frac{du}{dy}\right)=0 Standard form of a linear ODE The standard form of a second-order linear ODE is expressed with p, q and r known functions of x such that: $\boxed{y''+p(x)y'+q(x)y=r(x)}$ for which the total solution y is the sum of a homogeneous solution y_h and a particular solution y_p: $\boxed{y = y_h + y_p}$ Remark: if r=0, then the ODE is homogeneous (and we have y_p = 0). If r\neq0, then the ODE is said to be inhomogeneous. Linear dependency Two functions y_1, y_2 are said to be linearly dependent if \frac{y_2}{y_1}=C constant. Conversely, they are linearly independent if \frac{y_2}{y_1}\neq C. ## Linear homogeneous ### Variable coefficients Method of reduction of order Let y_1 be a solution to the equation y''+p(x)y'+q(x)y=0. By noting C_1, C_2 constants, the global solution y_h is written as: $\boxed{y_h=C_1y_1+C_2y_1\int\frac{e^{-\int pdx}}{y_1^2}dx}$ Remark: Here, for any function p, the notation \int pdx denotes the primitive of p without additive constant. ### Constant coefficients General form The general form of a linear homogeneous second-order ODE with a,b,c constant coefficients is: $\boxed{ay''+by'+cy=0}$ Resolution Based on the types of solution of the characteristic equation \boxed{a\lambda^2+b\lambda+c=0}, and by noting \boxed{\Delta=b^2-4ac} its discriminant, we distinguish the following cases:  Name Case Roots Solution Two distinct real roots \Delta>0 \displaystyle\lambda_1=\frac{-b+\sqrt{\Delta}}{2a}$$\displaystyle\lambda_2=\frac{-b-\sqrt{\Delta}}{2a}$ $y_h=C_1e^{\lambda_1x}+C_2e^{\lambda_2x}$ Double real root $\Delta=0$ $\displaystyle\lambda=-\frac{b}{2a}$ $y_h=[C_1+C_2x]e^{\lambda x}$ Complex conjugate roots $\Delta<0$ $\displaystyle\lambda_1=\alpha+i\beta$$\displaystyle\lambda_2=\alpha-i\betawhere \alpha=-\frac{b}{2a}and \beta=\frac{\sqrt{|\Delta|}}{2a} y_h=\left[C_1\cos(\beta x)+C_2\sin(\beta x)\right]e^{\alpha x} ### A special case: the Euler-Cauchy equation General form The Euler-Cauchy equation is a special case of linear homogeneous ODEs and has the following general form, where each a_i\in\mathbb{R} is a constant coefficient: $\boxed{a_nx^ny^{(n)}+a_{n-1}x^{n-1}y^{(n-1)}+...+a_1xy'+a_0y=0}$ Second-order case For n=2, by noting y=x^m, the ODE provides the indicial equation: $\boxed{am^2+(b-a)m+c=0}$ with discriminant \boxed{\Delta=(b-a)^2-4ac} and where the resolution of the ODE depends on the cases summarized in the table below.  Name Case Roots Solution Two distinct real roots \Delta>0 \displaystyle m_1=\frac{-b+a+\sqrt{\Delta}}{2a}$$\displaystyle m_2=\frac{-b+a-\sqrt{\Delta}}{2a}$ $y_h=C_1x^{m_1}+C_2x^{m_2}$ Double real root $\Delta=0$ $\displaystyle m=-\frac{b-a}{2a}$ $y_h=[C_1+C_2\ln|x|]x^{m}$ Complex conjugate roots $\Delta<0$ $\displaystyle m_1=\alpha+i\beta$$\displaystyle m_2=\alpha-i\betawhere \alpha=-\frac{b-a}{2a}and \beta=\frac{\sqrt{|\Delta|}}{2a} y_h=\left[C_1\cos(\beta \ln|x|)+C_2\sin(\beta \ln|x|)\right]x^{\alpha} ## Linear inhomogeneous ### Variable coefficients Wronskian Given y_1 and y_2 the two solutions of the homogeneous equation, we define the Wronskian W as follows: $\boxed{W=y_1y_2'-y_2y_1'}$ Method of Variation of Parameters The particular solution y_p of the inhomogeneous ODE is given by: $\boxed{y_p=-y_1\int\frac{y_2r}{W}dx+y_2\int\frac{y_1r}{W}dx}$ ### Constant coefficients Undetermined coefficients method The particular solution y_p of the inhomogeneous ODE ay'' + by' + cy = r(x) is determined from the correspondance table below:  Form of r Form of y_p C A x^n, n\in\mathbb{N}^* A_0+A_1x+...+A_nx^n e^{\gamma(x)} Ae^{\gamma x} \cos(\omega x) or \sin(\omega x) A\cos(\omega x)+B\sin(\omega x) x^ne^{\gamma x}\cos(\omega x) or x^ne^{\gamma x}\sin(\omega x) (A_0+A_1x+...+A_nx^n)\cos(\omega x)e^{\gamma x}+$$(B_0+B_1x+...+B_nx^n)\sin(\omega x)e^{\gamma x}$

Remark: all new constants are determined after plugging back $y_p$ into the ODE.

Modification rule If the particular solution $y_p$ picked from the above table matches either $y_1$ or $y_2$, then has to be multiplied by the lowest power of $x$ such that it is no more the case.

Sum rule If $r(x)$ is a sum of functions of the first column of the above table, then $y_p$ is the sum of its associated particular solutions.