Differential Equations Formula Sheet: Complete Reference Guide

Basics & Definitions

What is a Differential Equation?

A differential equation is an equation containing derivatives of a dependent variable \(y\) with respect to one or more independent variables.

Ordinary Differential Equations (ODE)

Equations with one independent variable.

\[\frac{dy}{dx} = f(x, y)\]

Partial Differential Equations (PDE)

Equations with two or more independent variables.

\[\frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2}\]

Order and Degree

Order: The order of a differential equation is the order of the highest derivative appearing in the equation.

Degree: The degree is the power of the highest order derivative when the equation is polynomial in derivatives.

                        Example: \(\left(\frac{d^2y}{dx^2}\right)^3 + 4\frac{dy}{dx} = x\) has order 2 and degree 3.
                    

Linearity

An ODE is linear if it can be written as:

\[a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \ldots + a_1(x)y' + a_0(x)y = b(x)\]

where \(y\) and all its derivatives appear to the first power only, and there are no products of \(y\) and its derivatives.

Homogeneous: If \(b(x) = 0\)
Nonhomogeneous: If \(b(x) \neq 0\)

First Order Differential Equations

Method 1 Separable Equations

If the equation can be written as:

\[\frac{dy}{dx} = \frac{A(x)}{B(y)}\]

Then separate variables and integrate:

\[\int B(y)\,dy = \int A(x)\,dx + C\]

Method 2 Homogeneous Equations

If \(M(x,y)dx + N(x,y)dy = 0\) where \(M\) and \(N\) are homogeneous of the same degree:

Substitution: Let \(y = vx\), so \(dy = v\,dx + x\,dv\)

Note: A function \(M(x,y)\) is homogeneous of degree \(n\) if \(M(cx, cy) = c^n M(x, y)\).

Method 3 Exact Equations

The equation \(M(x,y)dx + N(x,y)dy = 0\) is exact if:

\[\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\]

Solution: Find \(F(x,y)\) such that:

\[\frac{\partial F}{\partial x} = M \quad \text{and} \quad \frac{\partial F}{\partial y} = N\]

Then \(F(x,y) = C\) is the solution.

Method 4 Integrating Factors

If not exact, multiply by integrating factor \(\mu(x)\) or \(\mu(y)\):

\[\mu(x) = e^{\int \frac{1}{N}\left(\frac{\partial M}{\partial y} - \frac{\partial N}{\partial x}\right)dx}\]

(if the expression depends only on \(x\))

Method 5 Linear First Order

Standard form:

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

Integrating Factor: \(\mu(x) = e^{\int P(x)dx}\)

Solution:

\[y = \frac{1}{\mu(x)}\left[\int \mu(x)Q(x)dx + C\right]\]

Method 6 Bernoulli Equations

Form: \(\frac{dy}{dx} + P(x)y = Q(x)y^n\)

Substitution: Let \(v = y^{1-n}\), which transforms it into a linear equation.

Method 7 Clairaut's Equation

Form: \(y = xy' + f(y')\)

General Solution: \(y = Cx + f(C)\)

Singular Solution: Eliminate \(p = y'\) from \(y = xp + f(p)\) and \(\frac{dy}{dp} = 0\).

Second Order Linear Equations

General Form (Constant Coefficients)

\[ay'' + by' + cy = f(x)\]

where \(a, b, c\) are constants.

Case 1 Homogeneous Equations

When \(f(x) = 0\):

\[ay'' + by' + cy = 0\]

Characteristic Equation:

\[am^2 + bm + c = 0\]

Roots	General Solution
Real & Distinct: \(m_1, m_2\)	\(y = C_1e^{m_1x} + C_2e^{m_2x}\)
Real & Repeated: \(m_1 = m_2 = r\)	\(y = C_1e^{rx} + C_2xe^{rx}\)
Complex: \(\alpha \pm i\beta\)	\(y = e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x))\)

Case 2 Nonhomogeneous Equations

General solution:

\[y = y_h + y_p\]

where \(y_h\) is the homogeneous solution and \(y_p\) is a particular solution.

Method of Undetermined Coefficients

\(f(x)\)	Trial Solution \(y_p\)
\(ke^{ax}\)	\(Ae^{ax}\)
\(k\sin(ax)\) or \(k\cos(ax)\)	\(A\cos(ax) + B\sin(ax)\)
\(kx^n\)	\(A_nx^n + A_{n-1}x^{n-1} + \cdots + A_0\)
\(ke^{ax}\sin(bx)\)	\(e^{ax}(A\cos(bx) + B\sin(bx))\)

Note: If the trial solution is part of \(y_h\), multiply by \(x\) (or \(x^2\) if necessary).

Variation of Parameters

For \(y'' + p(x)y' + q(x)y = f(x)\), if \(y_1\) and \(y_2\) are solutions to the homogeneous equation:

\[y_p = u_1(x)y_1 + u_2(x)y_2\]

where:

\[u_1 = -\int \frac{y_2 f(x)}{W} dx, \quad u_2 = \int \frac{y_1 f(x)}{W} dx\]

and \(W = y_1y_2' - y_1'y_2\) is the Wronskian.

Euler-Cauchy Equation

Form: \(ax^2y'' + bxy' + cy = 0\)

Substitution: Try \(y = x^m\), leading to:

\[am(m-1) + bm + c = 0\]

Higher Order Linear Equations

nth Order Linear Homogeneous

\[a_ny^{(n)} + a_{n-1}y^{(n-1)} + \cdots + a_1y' + a_0y = 0\]

Characteristic Equation:

\[a_nm^n + a_{n-1}m^{n-1} + \cdots + a_1m + a_0 = 0\]

                        Solution Rules:
                        Real root \(m\): contributes \(Ce^{mx}\)
Repeated root \(m\) (k times): contributes \(C_1e^{mx} + C_2xe^{mx} + \cdots + C_kx^{k-1}e^{mx}\)
Complex roots \(\alpha \pm i\beta\): contribute \(e^{\alpha x}(C_1\cos(\beta x) + C_2\sin(\beta x))\)

                    

Systems of Differential Equations

Linear Systems

Form: \(\mathbf{x}' = A\mathbf{x}\) where \(A\) is a constant matrix.

Solution via Eigenvalues:

Find eigenvalues \(\lambda_i\) of matrix \(A\)
Find corresponding eigenvectors \(\mathbf{v}_i\)
Solution: \(\mathbf{x} = \sum C_i e^{\lambda_i t}\mathbf{v}_i\)

                        For 2×2 systems: If eigenvalues are \(\lambda_1, \lambda_2\) with eigenvectors \(\mathbf{v}_1, \mathbf{v}_2\):
                        \[\mathbf{x}(t) = C_1e^{\lambda_1 t}\mathbf{v}_1 + C_2e^{\lambda_2 t}\mathbf{v}_2\]
                    

Phase Plane Analysis

For autonomous systems \(\frac{dx}{dt} = f(x,y)\), \(\frac{dy}{dt} = g(x,y)\):

Critical Points: Solutions to \(f(x,y) = 0\) and \(g(x,y) = 0\)
Stability: Determined by eigenvalues of the Jacobian matrix at critical points

Laplace Transforms

Definition

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

Common Laplace Transforms

\(f(t)\)	\(\mathcal{L}\{f(t)\} = F(s)\)
\(1\)	\(\frac{1}{s}\)
\(t\)	\(\frac{1}{s^2}\)
\(t^n\)	\(\frac{n!}{s^{n+1}}\)
\(e^{at}\)	\(\frac{1}{s-a}\)
\(\sin(at)\)	\(\frac{a}{s^2+a^2}\)
\(\cos(at)\)	\(\frac{s}{s^2+a^2}\)
\(e^{at}\sin(bt)\)	\(\frac{b}{(s-a)^2+b^2}\)
\(e^{at}\cos(bt)\)	\(\frac{s-a}{(s-a)^2+b^2}\)

Properties

Property	Formula
Linearity	\(\mathcal{L}\{af + bg\} = a\mathcal{L}\{f\} + b\mathcal{L}\{g\}\)
First Derivative	\(\mathcal{L}\{f'\} = sF(s) - f(0)\)
Second Derivative	\(\mathcal{L}\{f''\} = s^2F(s) - sf(0) - f'(0)\)
nth Derivative	\(\mathcal{L}\{f^{(n)}\} = s^nF(s) - s^{n-1}f(0) - \cdots - f^{(n-1)}(0)\)
Integration	\(\mathcal{L}\left\{\int_0^t f(\tau)d\tau\right\} = \frac{F(s)}{s}\)
Translation (t-shift)	\(\mathcal{L}\{u(t-a)f(t-a)\} = e^{-as}F(s)\)
Translation (s-shift)	\(\mathcal{L}\{e^{at}f(t)\} = F(s-a)\)
Convolution	\(\mathcal{L}\{f * g\} = F(s)G(s)\)

Solving ODEs with Laplace Transforms

Take Laplace transform of both sides of the equation
Use initial conditions to simplify
Solve algebraically for \(Y(s) = \mathcal{L}\{y(t)\}\)
Use inverse Laplace transform to find \(y(t)\)

Tip: Use partial fraction decomposition for complex rational functions before taking inverse transform.

Series Solutions

Power Series Method

For equations of the form \(y'' + P(x)y' + Q(x)y = 0\), assume solution:

\[y = \sum_{n=0}^{\infty} a_n x^n\]

Substitute into the equation and match coefficients to find recurrence relation for \(a_n\).

Frobenius Method

For equations with regular singular point at \(x = 0\), assume:

\[y = x^r \sum_{n=0}^{\infty} a_n x^n = \sum_{n=0}^{\infty} a_n x^{n+r}\]

where \(r\) is determined by the indicial equation.

Indicial Equation:

Obtained by substituting the series and equating the coefficient of the lowest power of \(x\) to zero.

Special Functions

Bessel's Equation

\[x^2y'' + xy' + (x^2 - n^2)y = 0\]

Solutions: Bessel functions \(J_n(x)\) and \(Y_n(x)\)

Legendre's Equation

\[(1-x^2)y'' - 2xy' + n(n+1)y = 0\]

Solutions: Legendre polynomials \(P_n(x)\) and \(Q_n(x)\)

Hermite's Equation

\[y'' - 2xy' + 2ny = 0\]

Solutions: Hermite polynomials \(H_n(x)\)

Laguerre's Equation

\[xy'' + (1-x)y' + ny = 0\]

Solutions: Laguerre polynomials \(L_n(x)\)

Partial Differential Equations

Classification of Second-Order Linear PDEs

General form:

\[A\frac{\partial^2 u}{\partial x^2} + B\frac{\partial^2 u}{\partial x \partial y} + C\frac{\partial^2 u}{\partial y^2} + \text{lower order terms} = 0\]

Type	Condition	Example
Elliptic	\(B^2 - 4AC < 0\)	Laplace: \(\nabla^2 u = 0\)
Parabolic	\(B^2 - 4AC = 0\)	Heat: \(\frac{\partial u}{\partial t} = k\nabla^2 u\)
Hyperbolic	\(B^2 - 4AC > 0\)	Wave: \(\frac{\partial^2 u}{\partial t^2} = c^2\nabla^2 u\)

Heat Equation (Parabolic)

\[\frac{\partial u}{\partial t} = k\frac{\partial^2 u}{\partial x^2}\]

Physical Meaning: Describes heat diffusion in a material.

Solution Method: Separation of variables, Fourier series

Wave Equation (Hyperbolic)

\[\frac{\partial^2 u}{\partial t^2} = c^2\frac{\partial^2 u}{\partial x^2}\]

Physical Meaning: Describes wave propagation (vibrating string, sound waves).

d'Alembert Solution (infinite domain):

\[u(x,t) = \frac{1}{2}[f(x-ct) + f(x+ct)] + \frac{1}{2c}\int_{x-ct}^{x+ct} g(s)\,ds\]

Laplace Equation (Elliptic)

\[\nabla^2 u = \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} = 0\]

Physical Meaning: Steady-state heat distribution, electrostatic potential.

Solution Methods: Separation of variables, Green's functions, conformal mapping

Separation of Variables

Procedure:

Assume \(u(x,t) = X(x)T(t)\) (or similar for other variables)
Substitute into PDE and separate into two ODEs
Solve each ODE subject to boundary conditions
Combine solutions using superposition principle

Example for Heat Equation:

Assuming \(u(x,t) = X(x)T(t)\) leads to:

\[\frac{T'}{kT} = \frac{X''}{X} = -\lambda\]

This gives two ODEs: \(T' + k\lambda T = 0\) and \(X'' + \lambda X = 0\)

Fourier Series for PDEs

For boundary value problems on finite domains \([0, L]\):

\[u(x,t) = \sum_{n=1}^{\infty} B_n e^{-k\lambda_n t}\sin\left(\frac{n\pi x}{L}\right)\]

where coefficients \(B_n\) are determined from initial conditions using:

\[B_n = \frac{2}{L}\int_0^L f(x)\sin\left(\frac{n\pi x}{L}\right)dx\]

Numerical Methods

Euler's Method

For \(y' = f(x,y)\) with \(y(x_0) = y_0\):

\[y_{n+1} = y_n + hf(x_n, y_n)\]

where \(h\) is the step size.

Improved Euler (Heun's Method)

\[y_{n+1} = y_n + \frac{h}{2}[f(x_n, y_n) + f(x_{n+1}, y_n + hf(x_n, y_n))]\]

Runge-Kutta Method (4th Order)

\[\begin{align} k_1 &= hf(x_n, y_n)\\ k_2 &= hf(x_n + h/2, y_n + k_1/2)\\ k_3 &= hf(x_n + h/2, y_n + k_2/2)\\ k_4 &= hf(x_n + h, y_n + k_3)\\ y_{n+1} &= y_n + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4) \end{align}\]

Stability and Existence Theory

Existence and Uniqueness Theorem

For the IVP \(y' = f(x,y)\), \(y(x_0) = y_0\):

If \(f\) and \(\frac{\partial f}{\partial y}\) are continuous in a region containing \((x_0, y_0)\), then there exists a unique solution in some interval around \(x_0\).

Stability of Equilibrium Points

For autonomous system \(\frac{dy}{dx} = f(y)\), an equilibrium point \(y_e\) (where \(f(y_e) = 0\)) is:

Stable: if \(f'(y_e) < 0\)
Unstable: if \(f'(y_e) > 0\)
Semi-stable: if \(f'(y_e) = 0\)

Lyapunov Stability

An equilibrium point is stable if there exists a Lyapunov function \(V(x)\) such that:

\(V(0) = 0\) and \(V(x) > 0\) for \(x \neq 0\)
\(\frac{dV}{dt} \leq 0\) along trajectories

Applications

Population Growth

Exponential: \(\frac{dP}{dt} = kP\)

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

Radioactive Decay

\(\frac{dN}{dt} = -\lambda N\)

Solution: \(N(t) = N_0 e^{-\lambda t}\)

RLC Circuits

\(L\frac{d^2Q}{dt^2} + R\frac{dQ}{dt} + \frac{Q}{C} = E(t)\)

Spring-Mass Systems

\(m\frac{d^2x}{dt^2} + c\frac{dx}{dt} + kx = F(t)\)

Mixing Problems

\(\frac{dA}{dt} = \text{rate in} - \text{rate out}\)

Predator-Prey

Lotka-Volterra equations:

\(\frac{dx}{dt} = ax - bxy\)

\(\frac{dy}{dt} = -cy + dxy\)

Study Tips

Always check initial/boundary conditions before solving
Verify your solution by substituting back into the original equation
For linear equations, check if superposition principle applies
Draw phase portraits to understand system behavior
Use dimensional analysis to verify physical solutions
Remember: general solution = homogeneous solution + particular solution