My ODE

1st Order Differential Equations

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Definitions & Solutions

Representation

• Implicit: $F(x,y,y^{\prime },y^{\prime \prime },...y^{(n)})=0$
• Explicit: $y^{(n)}=f(x,y,y^{\prime },y^{\prime \prime },...,y^{(n-1)})$

Solutions of an ODE (function that verifies the equation)

• General Solution: $y_G(x)=...+C$ , $C\in \mathbb{R}$.
• Particular Solution: $y(x)$ → unique solution

Finding a General Solution

Direct Integration:

Form

If $y^{\prime }=f(x)$, then $y(x)=F(x)+C$, where $F(x)=\int f(x)dx$.
If $y^{\prime \prime }=f(x)$, then $y(x)=\int F(x)+C_{1}dx=F(F(x))+C_{1}x+C_2)$

Order $n$: The general solution of an order $n$ equation depends on $n$ parameters ($C_1,C_2,...C_n$).

Finding a Particular Solution

Those conditions need to be known:

• Initial Conditions: Values at a starting point ($y(x_0)=...$, $y^{\prime }(x_0)=...$, $y^{\prime \prime }(x_0)=...$ )
• Boundary Conditions: Set values at $n$ distinct points (e.g., $y(0)=2,y(2)=\mathrm{8...}$).

Separable Equations

Form

$y^{\prime }=\frac{g(x)}{h(y)}$

It cannot be integrated

Method:
Separate variables and integrate both sides:

Form

$\int h(y)dy=\int g(x)dx$

Substitutions

Goal: Transform $y^{\prime }=f(x,y)$ $\to$ simpler $z^{\prime }=g(x,z)$.

General Algorithm

1. Define $z$ (identify the “inner” function).
2. Differentiate to find $z^{\prime }$ (derivative w.r.t $x$).
3. Insert $z$ and $z^{\prime }$ back into original equation.

Common Patterns

Type	Original Form y′=	Substitution z=	Derivative z′=
Linear Arg	$f(ax+by+c)$	$ax+by+c$	$a+b\cdot y^{\prime }$
Homogeneous	$f\left(\frac{y}{x}\right)$	$\frac{y}{x}$	$\frac{y^{\prime }x-y}{x^2}$

Example from Notes

Given: $y^{\prime }=(4x+y{)}^2$

1. Sub: $z=4x+y$
2. Deriv: $z^{\prime }=4+y^{\prime }$ $⟹y^{\prime }=z^{\prime }-4$
3. Result: $z^{\prime }-4=z^2⟹\boxed{z^{\prime }=z^2+4}$

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Linear 1st Order DEs

Standard Form: $y^{\prime }=p(x)y+q(x)$
Homogeneous: If $q(x)=0$.
Inhomogeneous: If $q(x)\ne 0$.

Properties:
If $y_p$ is a particular solution to the inhomogeneous eq, and $y_h$ is the general solution to the homogeneous eq, then the total solution is:
$y(x)=y_h(x)+y_p(x)$

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Homogeneous Solution
For $y^{\prime }=p(x)y$:
Where $P(x)=\int p(x)dx$.

Form

$y_h=C\cdot e^{P(x)}$

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Inhomogeneous Solution
(Variation of Constants).
$P(x)$ is a primitive of $p(x)$.

Form

$y(x)=e^{P(x)}\left[\int q(x)e^{-P(x)}dx+C\right]$

Geometric Applications

Family of Curves

• Goal: Find the DE describing a family of curves (e.g., circles $x^2+y^2=r^2$).
• Method: Differentiate the equation and eliminate the parameter (e.g., $C$ or $r$).

Direction Fields

• For $y^{\prime }=f(x,y)$, the value $f(x,y)$ represents the slope of the solution curve at point $(x,y)$.
• Field lines: Curves tangent to the direction field vectors at every point.

Orthogonal Trajectories
Families of curves where every curve of one family intersects the other family at $90^{\circ }$.

• Method:
  1. Find DE of the original family: $y^{\prime }=f(x,y)$.
  2. The DE for orthogonal trajectories is:
     $y_{orth}^{\prime }=-\frac{1}{f(x,y)}$

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2nd Order Differential Equations

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Linear Homogeneous

Form: $y^{\prime \prime }+p{y}^{\prime }+qy=0$ (constant coeffs).
Ansatz: $y(x)=e^{\lambda x}$.
Characteristic Eq: ${\lambda }^2+p\lambda +q=0$.
Roots: ${\lambda }_{1,2}=\frac{-p\pm \sqrt{p^2-4q}}{2}$.

1. Over-damped ($p^2-4q>0$) Two real roots ${\lambda }_1,{\lambda }_2$. $y(x)=C_1{e}^{{\lambda }_{1}x}+C_2{e}^{{\lambda }_{2}x}$

2. Critically Damped ($p^2-4q=0$) Double root $\lambda =-p/2$. $y(x)=(C_1+C_{2}x)e^{\lambda x}$

3. Under-damped ($p^2-4q<0$) Complex roots ${\lambda }_{1,2}=\alpha \pm i\beta$. $\alpha =-p/2$, $\beta =\frac{\sqrt{4q-p^2}}{2}$. $y(x)=e^{\alpha x}(A\cos (\beta x)+B\sin (\beta x))$ Amplitude-phase: $y(x)=C{e}^{\alpha x}\cos (\beta x-\delta )$.

Superposition
If $y_1,y_2$ are independent solutions, $y=C_1{y}_1+C_2{y}_2$.

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Linear Inhomogeneous

Form: $y^{\prime \prime }+p{y}^{\prime }+qy=r(x)$.
Total Solution: $y(x)=y_h(x)+y_p(x)$.

Finding $y_p$ (Ansatz Method)
Based on perturbation function $r(x)$:

$r(x)$ Term	Ansatz $y_p(x)$
$e^{\lambda x}$	$A{e}^{\lambda x}$
$\cos (\omega x)$ / $\sin (\omega x)$	$A\cos (\omega x)+B\sin (\omega x)$
Polynomial $P_n(x)$	$C_n{x}^n+...+C_0$

Note: If Ansatz matches $y_h$, multiply by $x$ (or $x^2$).

“General Recipe” (from source)
Try: $y_p(x)=Ar(x)+B{r}^{\prime }(x)+C{r}^{\prime \prime }(x)$.
Substitute into DE and solve for A, B, C.

Reduction of Order

Transform $y^{\prime \prime }+p{y}^{\prime }+qy=r$ into a system:
Set $y_1=y,y_2=y^{\prime }$.
$\{\begin{array}{l}{y}_1^{\prime }=y_2\\ y_2^{\prime }=-q{y}_1-p{y}_2+r\end{array}$

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Numerical Methods

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Basics

Euler Method
$t_{i+1}=t_i+h$
$y_{i+1}=y_i+h\cdot f(t_i,y_i)$

Heun’s Method
(Improved Euler). Uses average of slopes at start and end of interval.

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Runge-Kutta 4 (RK4)

Uses weighted average of 4 slopes.

$k_1=f(t_i,y_i)$
$k_2=f(t_i+\frac{h}{2},y_i+\frac{h}{2}{k}_1)$
$k_3=f(t_i+\frac{h}{2},y_i+\frac{h}{2}{k}_2)$
$k_4=f(t_i+h,y_i+h{k}_3)$

$y_{i+1}=y_i+\frac{h}{6}(k_1+2k_2+2k_3+k_4)$