Differential Equations – Complete Reference Guide

1. First Order ODEs

Standard Form

$\frac{dy}{dx} = f(x, y)$ or $M(x,y)dx + N(x,y)dy = 0$

IVP (Initial Value)

$y(x_0) = y_0$ specifies a unique solution on interval containing $x_0$

Classification

Linear: $y' + P(x)y = Q(x)$

Separable: $\frac{dy}{dx} = f(x)g(y)$

Exact: $M_y = N_x$

Bernoulli: $y' + P(x)y = Q(x)y^n$

Homogeneous: $\frac{dy}{dx} = f(y/x)$

2. Separable Equations

Form & Solution

$\frac{dy}{dx} = g(x)h(y)$

Separate: $\int \frac{dy}{h(y)} = \int g(x) dx$

Steps

1. Rearrange to separable form

2. Integrate both sides

3. Solve for $y$ if possible

Example

$\frac{dy}{dx} = xy$

$\frac{dy}{y} = x dx$

$\ln|y| = \frac{x^2}{2} + C_1$

$y = Ce^{x^2/2}$

3. Linear 1st Order (Integrating Factor)

Standard Form

$y' + P(x)y = Q(x)$

Integrating Factor

$\mu(x) = e^{\int P(x) dx}$

Solution Steps

1. Identify $P(x)$ and $Q(x)$

2. Calculate $\mu(x)$

3. Multiply equation: $\mu y' + \mu P y = \mu Q$

4. $(\mu y)' = \mu Q$

5. $y = \frac{1}{\mu}\int \mu Q dx$

4. Exact Equations

Form & Exactness

$M(x,y)dx + N(x,y)dy = 0$

Exact if: $\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}$

Solution Method

1. Find $F(x,y)$: $\frac{\partial F}{\partial x} = M$

2. $F = \int M dx$ (treat $y$ constant)

3. Check: $\frac{\partial F}{\partial y} = N$

4. Solution: $F(x,y) = C$

Integrating Factor

If not exact, find $\mu$ making it exact

5. Bernoulli Equations

Form

$y' + P(x)y = Q(x)y^n$ $(n \neq 0,1)$

Solution Method

1. Substitute $v = y^{1-n}$

2. Then $v' = (1-n)y^{-n}y'$

3. Divide by $y^n$: $y^{-n}y' + Py^{1-n} = Q$

4. Linear in $v$: $v' + (1-n)Pv = (1-n)Q$

5. Solve for $v$, back-substitute

6. Substitution Methods

Homogeneous DE

$\frac{dy}{dx} = f(y/x)$

Let $v = y/x$, so $y = vx$

$\frac{dy}{dx} = v + x\frac{dv}{dx}$

Linear in $x$

$\frac{dx}{dy} + P(y)x = Q(y)$

Use integrating factor on $x$

Other Substitutions

$\frac{dy}{dx} = f(ax+by+c)$: $u = ax+by+c$

Reduce to separable form

7. 2nd Order Homogeneous

Form & Approach

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

Assume $y = e^{rx}$

Characteristic Eq

$ar^2 + br + c = 0$

Solve for Roots

$r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

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

8. Root Cases (2nd Order)

Distinct Real Roots

$r_1 \neq r_2$: $y = c_1 e^{r_1 x} + c_2 e^{r_2 x}$

Repeated Root

$r_1 = r_2 = r$: $y = c_1 e^{rx} + c_2 x e^{rx}$

Complex Roots

$r = \alpha \pm \beta i$

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

$\alpha$ = damping, $\beta$ = frequency

9. Reduction of Order

When Known Solution

$y_1(x)$ is solution to $y'' + P(x)y' + Q(x)y = 0$

Find $y_2$

1. Let $y_2 = v(x)y_1(x)$

2. $y_2' = v'y_1 + vy_1'$

3. $y_2'' = v''y_1 + 2v'y_1' + vy_1''$

4. Substitute into DE

5. Simplify to first-order in $w = v'$

Wronskian

$W = y_1 y_2' - y_1' y_2$

10. Undetermined Coefficients

Non-homogeneous: $ay'' + by' + cy = g(x)$

$y_{\text{gen}} = y_h + y_p$

Guess $y_p$ Form

$g(x) = P_n(x)$: try $y_p = x^s Q_n(x)$

$g(x) = e^{ax}$: try $y_p = x^s A e^{ax}$

$g(x) = \cos\beta x$: try $y_p = x^s(A\cos\beta x + B\sin\beta x)$

$s$ = multiplicity if $e^{ax}$ solves homogeneous

Solution Method

1. Compute $y_p'$ and $y_p''$

2. Substitute into DE

3. Solve for undetermined coefficients

11. Variation of Parameters

Formula for $y_p$

$y_p = -y_1 \int \frac{y_2 g(x)}{W} dx + y_2 \int \frac{y_1 g(x)}{W} dx$

Where

$W = y_1 y_2' - y_1' y_2$ (Wronskian)

$y_1, y_2$ are linearly independent solutions to homogeneous

$g(x)$ is forcing function from RHS

Alternative Form

$y_p = u_1(x)y_1 + u_2(x)y_2$

$u_1' = -y_2 g/W$, $u_2' = y_1 g/W$

12. Cauchy-Euler Equations

Form

$x^2 y'' + axy' + by = 0$ (equidimensional)

Solution Method

1. Assume $y = x^r$

2. $y' = rx^{r-1}$, $y'' = r(r-1)x^{r-2}$

3. Substitute to get characteristic equation:

$r(r-1) + ar + b = 0$

4. Solve for $r$

Solutions

Real $r_1, r_2$: $y = c_1 x^{r_1} + c_2 x^{r_2}$

Repeated $r$: $y = c_1 x^r + c_2 x^r \ln x$

Complex $\alpha \pm \beta i$: $y = x^\alpha(c_1\cos(\beta\ln x) + c_2\sin(\beta\ln x))$

13. Systems of Linear ODEs

Matrix Form

$\mathbf{x}' = A\mathbf{x}$ where $A$ is coefficient matrix

Solution Steps

1. Find eigenvalues: $\det(A - \lambda I) = 0$

2. Find eigenvectors: $(A - \lambda I)\mathbf{v} = 0$

3. Fundamental matrix: $\mathbf{X}(t) = e^{At}$

4. General solution: $\mathbf{x}(t) = c_1 e^{\lambda_1 t}\mathbf{v}_1 + c_2 e^{\lambda_2 t}\mathbf{v}_2$

Complex Eigenvalues

$\lambda = a \pm bi$: generate real-valued pair of solutions

14. Phase Portraits & Stability

Classification ($2 \times 2$ System)

Node: $\lambda_1, \lambda_2$ both real, same sign

Saddle: $\lambda_1 > 0$, $\lambda_2 < 0$ (different signs)

Focus/Spiral: Complex $\lambda = a \pm bi$ ($b \neq 0$)

Center: Pure imaginary $\lambda = \pm bi$

Stability

Stable (sink): $\text{Re}(\lambda) < 0$

Unstable (source): $\text{Re}(\lambda) > 0$

Marginally stable: $\text{Re}(\lambda) = 0$

15. Laplace Transforms - Definition

Definition

$\mathcal{L}\{f(t)\} = F(s) = \int_0^\infty e^{-st} f(t) dt$

Linearity

$\mathcal{L}\{af + bg\} = a\mathcal{L}\{f\} + b\mathcal{L}\{g\}$

Key Properties

$\mathcal{L}\{f'\} = sF(s) - f(0)$

$\mathcal{L}\{f''\} = s^2F(s) - sf(0) - f'(0)$

$\mathcal{L}\{e^{at}f(t)\} = F(s-a)$ (shift)

$\mathcal{L}\{f(t-a)u_a(t)\} = e^{-as}F(s)$ (delay)

16. Inverse Laplace & Common Pairs

Common Transform Pairs

$f(t)$	$F(s)$
$1$	$\frac{1}{s}$
$t^n$	$\frac{n!}{s^{n+1}}$
$e^{at}$	$\frac{1}{s-a}$
$\sin kt$	$\frac{k}{s^2+k^2}$
$\cos kt$	$\frac{s}{s^2+k^2}$

17. Solving IVPs with Laplace

General Procedure

1. Take Laplace of both sides of DE

2. Use $\mathcal{L}\{y'\} = sY - y(0)$ and $\mathcal{L}\{y''\} = s^2Y - sy(0) - y'(0)$

3. Substitute initial conditions

4. Solve for $Y(s)$

5. Apply $\mathcal{L}^{-1}$ to get $y(t)$

Partial Fractions

Often needed to decompose $Y(s)$ before inverse transform

18. Step & Impulse Functions

Unit Step (Heaviside)

$u_a(t) = \begin{cases} 0 & t < a \\ 1 & t \geq a \end{cases}$

$\mathcal{L}\{u_a(t)\} = \frac{e^{-as}}{s}$

Shifted Function

$\mathcal{L}\{f(t-a)u_a(t)\} = e^{-as}F(s)$

Dirac Delta ($\delta$-function)

$\int_{-\infty}^\infty \delta(t) dt = 1$

$\mathcal{L}\{\delta(t-a)\} = e^{-as}$

Impulse Response

Applied force at single instant

19. Series Solutions (Frobenius)

When to Use

$x^2 y'' + xp(x)y' + q(x)y = 0$ (singular point at $x=0$)

Frobenius Method

1. Assume $y = x^r \sum_{n=0}^\infty a_n x^n$

2. Find indicial equation: $r(r-1) + p_0 r + q_0 = 0$

3. For each root $r$, find recursion relation for $a_n$

4. Two solutions if $r_1 \neq r_2$

Regular vs Irregular

Regular singular pt: $xp(x), x^2q(x)$ analytic

20. Applications

Spring-Mass System

$m\ddot{x} + c\dot{x} + kx = F(t)$

$m$ = mass, $c$ = damping, $k$ = spring constant

RLC Circuit

$L\ddot{q} + R\dot{q} + \frac{1}{C}q = V(t)$

Analogous to mechanical system

Population Models

Logistic: $\frac{dP}{dt} = kP(1 - P/K)$

Predator-Prey: Lotka-Volterra system

Mixing Problems

$\frac{dA}{dt} = r_{\text{in}}c_{\text{in}} - r_{\text{out}}\frac{A}{V}$

DIFFERENTIAL EQUATIONS Complete Reference Sheet