Second Order Linear ODE Summary

Second Order Linear D.E. Summary

Homogenous D.E.
$y''+ay'+by=0$ .

Solve the Characteristic Equation: $\lambda^2+a\lambda+b=0$ .
Case 1) Two real roots $\lambda_1,\lambda_2$ : $\implies \boxed{y=c_1e^{\lambda_1x}+c_2e^{\lambda_2x}}$

Case 2) Real double root $\lambda$ : $\implies \boxed{y=c_1e^{\lambda x}+c_2xe^{\lambda x}}$

Case 3) Complex Conjugate root $\lambda_1,\lambda_2=-\frac{a}{2}\pm iw$ , where $w=\sqrt{b-\frac{a^2}{4}}$ : $\implies \boxed{y=e^{-\frac{a}{2}x}(c_1\cos wx+c_2\sin wx)}$

Non-homogenous D.E.
General solution of non-homogenous D.E.: $\displaystyle y=y_h+y_p,$ where $y_h$ is the general solution of the homogenous equation, and $y_p$ is the particular solution (with no arbitrary constants).

Method of Undetermined Coefficients (Guess and try method)
$y''+p(x)y'+q(x)y=r(x)$ .

Only works if $r(x)$ is polynomial, exponential, sine or cosine (or sum/product of these).

Polynomial: Try $y$ =Polynomial (e.g. $y=Ax^2+Bx+C$ or $y=Bx+C$ .)

Exponential ( $e^{kx}$ ): Try $y=ue^{kx}$ , where $u$ is a function of $x$ .

Trigonometric ( $\sin kx$ or $\cos kx$ ): Convert to complex differential equation by replacing $y$ with $z$ , replace $\sin kx$ / $\cos kx$ by $e^{ikx}$ .

Try $z=ue^{ikx}$ , where $u$ is a function of $x$ . After solving for $z$ , take real/imaginary part of $z$ for cosine/sine respectively.

Method of variation of parameters
$y''+p(x)y'+q(x)y=r(x)$ .

[Step 1)] Solve the homogenous D.E. $y''+p(x)y'+q(x)y=0$ .

Get solution of the form $y_h=c_1y_1+c_2y_2$ .

[Step 2)]
Let $\displaystyle u=-\int\frac{y_2r}{W}\,dx$ and $\displaystyle v=\int\frac{y_1r}{W}\,dx$ where $W$ is the Wronskian $\displaystyle W=y_1y_2'-y_1'y_2.$

Particular solution: $y_p=uy_1+vy_2$ .

General solution: $y=y_h+y_p$ .

Forced Oscillations
Let $F_0$ be the amplitude of the driving (external) force. If $F_0=0$ , by Newton’s Second Law, $m\ddot{x}=-kx$ , hence $\displaystyle \boxed{\ddot{x}=-\omega^2 x},$
where $\omega=\sqrt{k/m}$ . The value $\omega$ is called the natural frequency.