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

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
 * 3) Real and distinct, then $$y(x)=Ae^{\lambda_1 x}+Be^{\lambda_2 x}$$
 * 4) Real and equal, then $$y(x)=(Ax+B)e^{\lambda_1 x}$$
 * 5) Imaginary, $$\lambda_i=a\pm bi$$, then $$y(x)=(A\cos{bx}+B\sin{bx})e^{a x}$$

General Form
$$ax^2y''+bxy'+cy=0$$ or $$p(D)y=0$$ where
 * $$p(D)=ax^2D^2+bxD+c$$ is called the polynomial differential operator.

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

If one particular solution is known
If one solution of a homogeneous linear second order equation is known, $$y_1 (x)\ne0$$, 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 $$W(x)=0,\forall x$$.