Differential Equations

1. Linearly dependent functions. If $f(x)=cg(x)$, where $c$ is a constant, then they are linearly dependent.

2. If $y_1(x),y_2(x)$ are solutions to $P(x)y”+Q(x)y’+R(x)y=0$, then $y(x)=y_1(x)+y_2(x)$ is also a solution.

3. Solve $ay”+by’+cy’=0$

First, solve the characteristic equation:

$$ar^2+br+c=0$$

$$\Delta = b^2-4ac$$

If $\Delta>0$:

$$y(x)=c_1e^{r_1x}+c_2e^{r_2x}$$

If $\Delta=0$, $r=\frac{-b}{2a}$:

$$y(x)=c_1e^{rx}+c_2xe^{rx}$$

If $\Delta<0$, $r=\alpha \pm \beta i$

$$y(x)=e^{\alpha x}(c_1\cos\beta x+c_2\sin\beta x)$$

If the value of $y(x_0)$ and $y'(x_0)$ are given, the solution is unique. This is called the initial value problem.

If the function values at different $x$, the solution might not exist, uniquely exist, or exist infinitely many. This is called the boundary value problem.

5. Solve $ay”+by’+cy=G(x)$

The general solution is: $y(x)=y_p(x)+y_c(x)$

$y_c(x)$ is the general solution to the complementary equation $ay”+by’+cy’=0$

The particular solution $y_p(x)$ depends the type of function $G(x)$ is

Type 1: polynomial

Let $y_p(x)$ be a polynomial with the same order as $G(x)$ .

Type 2: $G(x)=Ce^{kx}G(x)=Ce^{kx}$

Let $y_p(x)=Ae^{kx}$

Type 3: $G(x)=C\cos (kx)$ or $C\sin (kx)$

Let $y_p(x)=A\cos (kx)+B\sin (kx)$

Type 4: $G(x)=G_1(x)+G_2(x)$

$y_p(x)=y_{p1}(x)+y_{p2}(x)$. $y_{p1}(x)$ is the particular solution for $ay”+by’+cy=G_1(x)$

Type 5: $G(x)=G_1(x)G_2(x)$

Let $y_p(x)$ be the product of the types of functions. For example, if $G(x)=(2x+1)e^{3x}$ , let $y_p(x)=(Ax+B)e^{3x}$. Note that $y_p(x)\neq y_{p1}(x)y_{p2}(x)$.

If $G(x)=P(x)\sin(kx)$ or $P(x)\cos(kx)$, where $P(x)$ is a polynomial, let $y_p(x)=P_1(x)\cos (kx)+P_2(x)\sin(kx)$. For example, $G(x)=(2x+1)\sin3x$. Let $y_p(x)=(Ax+B)\cos3x+(Cx+D)\sin3x$

Once the type of function is decided, plug $y_p(x)$ in and solve for the coefficients.

If the particular solution has the same form as the complementary solution, multiply $y_p(x)$ by $x$.

For example, $-y”+5y’-6y=e^{2x}$. $y_c(x)=c_1e^{2x}+c_2e^{3x}$. If we follow the rules we should let y_p(x)=Ae^{2x}, but since there is $e^{2x}$ in it, we should let $y_p(x)=Axe^{2x}$

7. Solve $P(x)y”+Q(x)y’+R(x)y=0$, where $P(x),Q(x),R(x)$ are polynomials.

Let

$$y=\sum^{\infty}_{n=0}C_nx^n$$

$$y’=\sum^{\infty}_{n=0}(n+1)C_{n+1}x^n$$

$$y”=\sum^{\infty}_{n=0}(n+2)(n+1)C_{n+2}x^n$$

8. Spring equations

Undamped spring:

$$mx”+kx=0$$

$$x(t)=A\cos(\omega t+\delta), \omega=\sqrt\frac k m$$

Damped spring:

$$mx”+cx’+kx=0$$

Driven Oscillation:

$$mx”+cx’+kx=F(t)$$

9. RLC circuit

$$LQ”+RQ+\frac 1 CQ=E(t)$$