Differential Equation Calculator

About Differential Equations

A differential equation is an equation that relates one or more functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two.

Types of Differential Equations

There are several types of differential equations:

Ordinary Differential Equations (ODEs):

These involve functions of a single variable and their derivatives. Our calculator focuses on ODEs.

Partial Differential Equations (PDEs):

These involve functions of multiple variables and their partial derivatives.

First Order ODEs

First order differential equations are equations involving only the first derivative of the unknown function. Common types include:

• Separable equations: Can be written as \( f(y)dy = g(x)dx \)
• Exact equations: \( M(x,y)dx + N(x,y)dy = 0 \) where \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \)
• Linear equations: \( \frac{dy}{dx} + P(x)y = Q(x) \)
• Homogeneous equations: Can be written as \( \frac{dy}{dx} = f\left(\frac{y}{x}\right) \)
• Bernoulli equations: \( \frac{dy}{dx} + P(x)y = Q(x)y^n \)

Second Order ODEs

Second order differential equations involve the second derivative of the unknown function. Common types include:

• Linear homogeneous with constant coefficients: \( ay'' + by' + cy = 0 \)
• Linear nonhomogeneous with constant coefficients: \( ay'' + by' + cy = f(x) \)
• Euler-Cauchy equations: \( x^2y'' + pxy' + qy = 0 \)

Applications

Differential equations are used to model many real-world phenomena including:

• Population growth and decay
• Newton's law of cooling
• Simple harmonic motion
• Electrical circuits
• Fluid dynamics
• Chemical reactions