Ordinary Differential Equations:Cheat Sheet/Second Order Homogeneous Ordinary Differential Equations

With Constant Coefficients

General Form

$a{y}^{″}+b{y}^{\prime }+cy=0$ or $p(D)y=0$, where

$p(D)=a{D}^2+bD+c$ is called the polynomial differential operator with constant coefficients.

Solution

1. Solve the auxiliary equation, $p(m)=0$, to get $m={\lambda }_1,{\lambda }_2$
2. If ${\lambda }_1,{\lambda }_2$ are
   1. Real and distinct, then $y(x)=A{e}^{{\lambda }_{1}x}+B{e}^{{\lambda }_{2}x}$
   2. Real and equal, then $y(x)=(Ax+B)e^{{\lambda }_{1}x}$
   3. Imaginary, ${\lambda }_i=a\pm bi$, then $y(x)=(A\cos bx+B\sin bx)e^{ax}$

Euler-Cauchy Equations

General Form

$a{x}^2{y}^{″}+bx{y}^{\prime }+cy=0$ or $p(D)y=0$ where

$p(D)=a{x}^2{D}^2+bxD+c$ is called the polynomial differential operator.

Solution

Solving $a{x}^2{y}^{″}+bx{y}^{\prime }+cy=0$ is equivalent to solving $a{y}^{″}+(b-a)y^{\prime }+cy=0$

General Homogenous ODE with Variable Coefficients

If one particular solution is known

If one solution of a homogeneous linear second order equation is known, $y_1(x)\ne 0$, original equation can be converted to a linear first order equation using substitutions $y_2=y_2(x)z(x)$ and subsequent replacement $z^{\text{'}}(x)=u$.

Abel's identity

For the homogeneous linear ODE $y^{″}+p(x)y^{\prime }+q(x)y=0$, Wronskian of its two solutions is given by $W_{(}{y}_1,y_2)(x)=W(x_0)e^{-\underset{x_0}{\overset{x}{\int }}p(x)dx}$

Solution with Abel's identity

Given a homogenous linear ODE and a solution of ODE, $y_1(x)$, find Wronskian using Abel’s identity and by definition of Wronskian, equate and solve for $y_2(x)$.

Few Useful Notes

1. If $y_1,y_2$ are linearly dependent, $W(x)=0,\mathrm{\forall }x$
2. If $W(x)=0$, for some $x$, then $W(x)=0,\mathrm{\forall }x$.