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:

1. Bring the $y$ & $dy$ to one side and $x$ & $dx$ to the other side like shown above (seperation)
2. Integrate both sides (don't forget to add a constant $C$)
3. 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 $k$:

1. Find a single differential euqation that statisfies all members of the familiy
• Elimitate $k$ by substituion
2. The differential equation for the orthogonal trajectories is $-1/y'$
3. 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:

1. If step 2 cannot be applied directly: multiply both sides by the integrating factor: $e^{P(x)}$
2. Apply the inverse product rule on left side
3. Integrate both sides (don't forget to add a constant $C$)
4. 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: $ay'' + by' + cy = 0$

1. Calculate the complementary solution $y_c$
2. Find a particular solution $y_p$
3. General solution is $y = y_c + y_p$