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

With Constant Coefficients

General Form

$ay''+by'+cy=0$ or $p(D)y=0$, where

$p(D)=aD^{2}+bD+c$ 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 $y(x)=Ae^{\lambda _{1}x}+Be^{\lambda _{2}x}$
   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

$ax^{2}y''+bxy'+cy=0$ or $p(D)y=0$ where

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

Solution

Solving $ax^{2}y''+bxy'+cy=0$ is equivalent to solving $ay''+(b-a)y'+cy=0$

General Homogenous ODE with Variable Coefficients

If one particular solution is known

If one solution of a homogeneous linear second order equation is known, $y_{1}(x)\neq 0$, original equation can be converted to a linear first order equation using substitutions $y_{2}=y_{2}(x)z(x)$ and subsequent replacement $z^{'}(x)=u$.

Abel's identity

For the homogeneous linear ODE $y''+p(x)y'+q(x)y=0$, Wronskian of its two solutions is given by $W_{(}y_{1},y_{2})(x)=W(x_{0})e^{-\int _{x_{0}}^{x}p(x)dx}$

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, $W(x)=0,\forall x$
2. If $W(x)=0$, for some $x$, then $W(x)=0,\forall x$.