1 second order linear differential equations

1.1 homogenous

ay'' + by' + cy = 0

1. y = e^{rx}
2. ar^2e^{rx} + bre^{rx} + ce^{rx} = 0
3. ar^2 + br + c = 0
4. y = e^{r_1x}\ \lor\ y = e^{r_2x}
5. for c_1, c_2 as arbitrary constants: y = c_1e^{r_1x} + c_2e^{r_2x}

• If r is complex, the general solution is in the form: y = e^{Re(r)x}(b_1 \cos(Im(r)x) + b_2 \sin(Im(r)x))
• If r_1 = r_2 then y = c_1e^{rx} + c_2xe^{rx}

1.2 non-homogenous

ay'' + by' + cy = g(x)

Let y_h be the solution to ay'' + by' + cy = 0 and y_p is a collection of functions that differentiated give g(x)

Then y = y_h + y_p

1.2.1 method of undetermined coefficients

Examples:

1. ay'' + by' + cy = \sin(x)
2. y_p = A\cos(x) + B\sin(x)
   • y_p' = -A\sin(x) + B\cos(x)
   • y_p'' = -A\cos(x) - B\sin(x)
3. a(-A\cos(x) - B\sin(x)) + b(-A\sin(x) + B\cos(x)) + c(A\cos(x) + B\sin(x)) = \sin(x)
4. (-Aa + Bb + Ac)\cos(x) + (-Ba - Ab + Bc)\sin(x) = \sin(x)
5. So: -Aa + Bb + Ac = 0 and -Ba - Ab + Bc = 1
6. Solving step 5 we get A and B

1.2.2 method of variation of parameters

y_p = v_1 y_1 + v_2 y_2

\begin{cases} v_1'y_1 + v_2'y_2 = 0 \\ v_1'y_1' + v_2'y_2' = \frac{g(x)}{a} \\ \end{cases}

1.2.2.1 Wronskian

W(x) = y_1y_2' + y_1'y_2

Then: v_1 = -\int \frac{g(x)y_2}{a \cdot W(x)} dx and v_2 = \int \frac{g(x)y_1}{a \cdot W(x)} dx

1.3 damped harmonic motion

\frac{d^2x}{dt^2} + \beta\frac{dx}{dt} + \omega^2x = 0 where \beta = \frac{\gamma}{m} and \omega^2 = \frac{k}{m}

We can see this as a differential equation:

r^2 + \beta r + \omega^2 = 0

• if \beta^2 > 4\omega^2 (two real roots) we call it overdamped
• if \beta^2 = 4\omega^2 (one real root) we call it critically damped
• if \beta^2 < 4\omega^2 (complex roots) we call it an underdamped case
  • let \overline \omega = \omega \sqrt{1 - \frac{\beta^2}{4\omega^2}}
  • then x = e^\frac{-\beta t}{2}(c_1 \cos\overline\omega t + c_2 \sin\overline\omega t)

1.4 damped forced oscillations

m\frac{d^2x}{dt^2} + \gamma \frac{dx}{dt} + kx = F_0 \cos \Omega t

The solution is:

x(t) = c_1e^{r_1t} + c_2e^{r_2t} + \frac{F_0}{\sqrt{m^2(\omega^2 - \Omega^2)^2 + \gamma^2\Omega^2}}\cos(\Omega t - \delta)

Where \delta = \tan^{-1}\big (\frac{\gamma \Omega}{m(\omega^2 - \Omega^2)}\big )