Differential Equation Solver

A differential equation (DE) is an equation that relates a function to its derivatives. An ordinary differential equation (ODE) involves a function of one variable:

$F\left(x, y, y', y'', \ldots, y^{(n)}\right) = 0$

The order of a DE is the highest derivative that appears. The degree is the power of the highest-order derivative (when the equation is polynomial in derivatives).

First-order ODE: $y' = f(x, y)$

Second-order ODE: $y'' + p(x)y' + q(x)y = g(x)$

A solution is a function $y(x)$ that satisfies the equation on some interval. The general solution contains arbitrary constants (one per order). An initial value problem (IVP) specifies conditions like $y(x_0) = y_0$ to determine a unique particular solution.

Differential equations model real-world phenomena: population growth, radioactive decay, spring-mass systems, electrical circuits, heat conduction, and fluid flow.

Method 1: Separation of Variables

For equations of the form $\frac{dy}{dx} = f(x)g(y)$:

1. Separate: $\frac{dy}{g(y)} = f(x)\,dx$
2. Integrate both sides: $\int \frac{dy}{g(y)} = \int f(x)\,dx$

Example: $\frac{dy}{dx} = 2xy$ → $\frac{dy}{y} = 2x\,dx$ → $\ln|y| = x^2 + C$ → $y = Ae^{x^2}$

Method 2: Integrating Factor (First-Order Linear)

For $y' + P(x)y = Q(x)$, multiply by the integrating factor $\mu(x) = e^{\int P(x)\,dx}$:

$\frac{d}{dx}[\mu(x) \cdot y] = \mu(x) \cdot Q(x)$

Then integrate both sides to find $y$.

Example: $y' + 2y = e^{-x}$. Here $P(x) = 2$, so $\mu = e^{2x}$. Multiply: $(e^{2x}y)' = e^{x}$. Integrate: $e^{2x}y = e^x + C$, so $y = e^{-x} + Ce^{-2x}$.

Method 3: Characteristic Equation (Constant Coefficients)

For $ay'' + by' + cy = 0$, solve the characteristic equation $ar^2 + br + c = 0$:

Method 4: Undetermined Coefficients

For $ay'' + by' + cy = g(x)$ where $g(x)$ is a polynomial, exponential, sine, cosine, or combination:

1. Find the general solution to the homogeneous equation
2. Guess a particular solution form based on $g(x)$
3. Substitute and solve for coefficients
4. General solution = homogeneous + particular

Method 5: Variation of Parameters

A general method for $y'' + p(x)y' + q(x)y = g(x)$ when the homogeneous solutions $y_1, y_2$ are known:

$y_p = -y_1 \int \frac{y_2 g}{W}\,dx + y_2 \int \frac{y_1 g}{W}\,dx$

where $W = y_1 y_2' - y_2 y_1'$ is the Wronskian.

Comparison of Methods

• Forgetting the constant of integration: In separation of variables, the constant must be included before solving for $y$, as it affects the final form of the solution.
• Incorrect integrating factor: The integrating factor for $y' + P(x)y = Q(x)$ is $e^{\int P(x)\,dx}$. Make sure the equation is in standard form (coefficient of $y'$ must be 1) before identifying $P(x)$.
• Missing the repeated-root case: When the characteristic equation has a repeated root $r$, the second solution is $xe^{rx}$, not just $e^{rx}$ again.
• Wrong particular solution guess: If your guess for $y_p$ is already a solution to the homogeneous equation, multiply by $x$ (or $x^2$ if needed) to get a valid form.
• Ignoring initial conditions: The general solution has arbitrary constants. Apply initial conditions only after finding the complete general solution.