Differential Equations Formulas
First-Order Differential Equations
Linear First-Order Differential Equation: \[ \frac{dy}{dx} + P(x)y = Q(x) \]
Solution of Linear First-Order Differential Equation: \[ y(x) = e^{-\int P(x) , dx} \left( \int Q(x) e^{\int P(x) , dx} , dx + C \right) \]
Separable Differential Equation: \[ \frac{dy}{dx} = g(y)h(x) \]
Solution of Separable Differential Equation: \[ \int \frac{1}{g(y)} , dy = \int h(x) , dx + C \]
Exact Differential Equation: \[ M(x, y) , dx + N(x, y) , dy = 0 \]
Condition for Exactness: \[ \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \]
Integrating Factor (for non-exact equations): \[ \mu(x) = e^{\int P(x) , dx} \] \[ \mu(y) = e^{\int Q(y) , dy} \]
Second-Order Differential Equations
Homogeneous Linear Second-Order Differential Equation: \[ a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + c y = 0 \]
Characteristic Equation: \[ ar^2 + br + c = 0 \]
Solution for Distinct Real Roots ((r_1) and (r_2)): \[ y(x) = C_1 e^{r_1 x} + C_2 e^{r_2 x} \]
Solution for Repeated Roots ((r)): \[ y(x) = (C_1 + C_2 x)e^{rx} \]
Solution for Complex Roots ((\alpha \pm \beta i)): \[ y(x) = e^{\alpha x} (C_1 \cos(\beta x) + C_2 \sin(\beta x)) \]
Non-Homogeneous Linear Second-Order Differential Equation: \[ a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + c y = f(x) \]
General Solution: \[ y(x) = y_h(x) + y_p(x) \]
- (y_h(x)) is the solution to the homogeneous equation.
- (y_p(x)) is a particular solution to the non-homogeneous equation.
Method of Undetermined Coefficients: Assume a form for (y_p(x)) based on (f(x)) and solve for coefficients.
Variation of Parameters: \[ y_p(x) = y_1(x) \int \frac{y_2(x) f(x)}{W(y_1, y_2)} , dx - y_2(x) \int \frac{y_1(x) f(x)}{W(y_1, y_2)} , dx \] where (y_1(x)) and (y_2(x)) are solutions to the corresponding homogeneous equation and (W(y_1, y_2)) is the Wronskian.
Higher-Order Differential Equations
General Linear (n)th-Order Differential Equation: \[ a_n \frac{d^n y}{dx^n} + a_{n-1} \frac{d^{n-1} y}{dx^{n-1}} + \cdots + a_1 \frac{dy}{dx} + a_0 y = f(x) \]
Solution: \[ y(x) = y_h(x) + y_p(x) \]
Systems of Differential Equations
First-Order Linear System: \[ \mathbf{X}' = A \mathbf{X} + \mathbf{B} \]
Solution Using Eigenvalues and Eigenvectors:
- Find eigenvalues (\lambda) and eigenvectors (\mathbf{v}) of matrix (A).
- General solution: \[ \mathbf{X}(t) = c_1 e^{\lambda_1 t} \mathbf{v}_1 + c_2 e^{\lambda_2 t} \mathbf{v}_2 + \cdots + c_n e^{\lambda_n t} \mathbf{v}_n \]
Phase Plane Analysis:
- Classify critical points (nodes, saddles, spirals, centers) based on eigenvalues.
Laplace Transforms
Definition: \[ \mathcal{L}{f(t)} = \int_0^\infty e^{-st} f(t) , dt \]
Inverse Laplace Transform: \[ \mathcal{L}^{-1}{F(s)} = \frac{1}{2\pi i} \int_{\gamma - i\infty}^{\gamma + i\infty} e^{st} F(s) , ds \]
Properties:
- Linearity: \[ \mathcal{L}{af(t) + bg(t)} = a\mathcal{L}{f(t)} + b\mathcal{L}{g(t)} \]
- First Derivative: \[ \mathcal{L}{f'(t)} = sF(s) - f(0) \]
- Second Derivative: \[ \mathcal{L}{f''(t)} = s^2 F(s) - sf(0) - f'(0) \]
Laplace Transform of Common Functions: \[ \mathcal{L}{1} = \frac{1}{s} \] \[ \mathcal{L}{t^n} = \frac{n!}{s^{n+1}} \] \[ \mathcal{L}{e^{at}} = \frac{1}{s-a} \] \[ \mathcal{L}{\sin(at)} = \frac{a}{s^2 + a^2} \] \[ \mathcal{L}{\cos(at)} = \frac{s}{s^2 + a^2} \]
Fourier Series
Fourier Series Representation: \[ f(x) = a_0 + \sum_{n=1}^{\infty} \left( a_n \cos \frac{n\pi x}{L} + b_n \sin \frac{n\pi x}{L} \right) \]
Coefficients: \[ a_0 = \frac{1}{2L} \int_{-L}^{L} f(x) , dx \] \[ a_n = \frac{1}{L} \int_{-L}^{L} f(x) \cos \frac{n\pi x}{L} , dx \] \[ b_n = \frac{1}{L} \int_{-L}^{L} f(x) \sin \frac{n\pi x}{L} , dx \]
Partial Differential Equations
Heat Equation: \[ \frac{\partial u}{\partial t} = \alpha \frac{\partial^2 u}{\partial x^2} \]
Wave Equation: \[ \frac{\partial^2 u}{\partial t^2} = c^2 \frac{\partial^2 u}{\partial x^2} \]
Laplace's Equation: \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0 \]
Poisson's Equation: \[ \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = f(x, y) \]
Boundary and Initial Conditions
Initial Conditions:
- (y(0) = y_0)
- (y'(0) = y_1)
Boundary Conditions:
- Dirichlet: (u(a, t) = 0)
- Neumann: (\frac{\partial u}{\partial x} \bigg|_{x=a} = 0)
Separation of Variables: Assume (u(x,t) = X(x)T(t)) and separate the PDE into ODEs for (X(x)) and (T(t)).