## 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.

If $F_0\neq 0$, then $\displaystyle \boxed{m\ddot{x}+kx=F_0\cos\alpha t},$
where $\alpha$ is the driving (external) frequency.

At resonance (when $\alpha=\omega$), $\displaystyle \boxed{x=\frac{F_0t}{2m\omega}\sin(\omega t)}.$ 