# Differential Equations

**Differential equation**: an equation that involves a function and its diravatives

**General solution**: solution for a differential euqation with constants that holds for all possible functions

**Particular solution**: solution for a differential equation on a particular coordinate

**Initial-value problem**: fill in $x, y$ in a general solution to get the value of constants $c_n$ to get the particular solution

## Seperable equations #

**Seperable differential equation**: differential equations in the form: $$ \frac{dy}{dx} = g(x)f(y) $$

Which can be rewritten as: (where the x's and y's are seperated to solve it) $$ h(y) dy = g(x) dx $$

**Solving seperable differential equations**:

- Bring the $y$ & $dy$ to one side and $x$ & $dx$ to the other side like shown above (seperation)
- Integrate both sides (don't forget to add a constant $C$)
- Rewrite to $y = ...$

### Orthogonal trajectories #

**Orthogonal trajectory**: a curve that intersects each curve of a **famility of curves** orthogonally

To find orthogonal trajectories of a famility with variable $k$:

- Find a single differential euqation that statisfies all members of the familiy
- Elimitate $k$ by substituion

- The differential equation for the orthogonal trajectories is $-1/y'$
- Solve the differential equation to find a famility of functions for the orthogonal trajectories

## First-order linear equations #

**First-order linear differential equations**: are equations in the form: $$ \frac{dy}{dx} + p(x)y = q(x) $$

Solving first-order linear differential equations:

- If step 2 cannot be applied directly: multiply both sides by the integrating factor: $e^{P(x)}$
- Apply the
*inverse*product rule on left side - Integrate both sides (don't forget to add a constant $C$)
- Rewrite to $y = ...$

## Second-order linear equations #

**Second-order linear differential equations**: are equations in the form:

$$ ay'' + by' + cy = f(x) $$

- If $f(x) = 0$ equation is
**homogeneous** - If $f(x) \neq 0$ the equation is
**non-homogeneous**

### Homogeneous second-order linear DEs #

$$ ay'' + by' + cy = 0 $$

- Two district roots $b^ - 4ac > 0$: $$ y = c_1 e^{r_1 x} + c_2 e^{r_2 x} $$
- Identical roots $b^2 - 4ac = 0$: $$ y = c_1 e^{r x} + c_2 x e^{r x} = (c_1 + c_2 x)e^{r x} $$

### Non-homogeneous second-order linear DEs #

$$ ay'' + by' + cy = f(x) $$ Has the **complementary euqation**: $ay'' + by' + cy = 0$

- Calculate the complementary solution $y_c$
- Find a particular solution $y_p$
- General solution is $y = y_c + y_p$