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Differential Equation Calculator

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Differential Equation Solver

Our Differential Equation Solver helps you quickly solve equations involving functions and their derivatives. It supports first-order, second-order, and higher-order differential equations, providing clear, step-by-step solutions. You can enter equations manually, select from common templates, or even upload an image for automatic processing.

What Is a Differential Equation?

A differential equation is an equation that connects a function with its derivatives.

General Form:

\( F\left(x, y, \frac{dy}{dx}, \frac{d^2y}{dx^2}, \dots, \frac{d^n y}{dx^n}\right) = 0 \)

Explanation:

Differential equations describe how a quantity changes with respect to one or more variables. The function represents the quantity of interest, while its derivatives indicate how the quantity evolves over time or space.

Steps to Solve a Differential Equation

Step 1 – Simplify the equation: Rearrange terms to isolate variables.
Step 2 – Integrate: Apply integration to both sides, including constants of integration as needed.
Step 3 – Solve for the unknown function: Express the dependent variable explicitly.
Step 4 – Apply initial conditions: Use given values to determine constants.

Example:

Solve: dy/dx = 4y

Solution:

Step 1 – Identify type: This is a first-order linear differential equation.

Step 2 – Separate variables: 1/y dy = 4 dx

Step 3 – Integrate both sides: ∫ 1/y dy = ∫ 4 dx

ln |y| = 4x + C

Step 4 – Solve for y: y = e^(4x + C) = C e^(4x)

Step 5 – Apply initial condition: if y(0) = 3

3 = C e^0 ⟹ C = 3

Final Solution: y = 3 e^(4x)

Using the Differential Equation Solver

Type or paste the equation into the input field.
Optionally, upload an image of the equation.
Click “Solve” to get the step-by-step solution instantly.

The solver provides both general solutions and allows you to find particular solutions using initial conditions.

Types of Differential Equations

Ordinary Differential Equations (ODEs)

Involve derivatives with respect to a single independent variable.

Example: dy/dx + 2y = x

Partial Differential Equations (PDEs)

Involve partial derivatives with respect to multiple independent variables.

Example: ∂u/∂t = k ∂²u/∂x²

Linear Differential Equations

The dependent variable and its derivatives appear only to the first power and are not multiplied together.

Example: a_2(x) d²y/dx² + a_1(x) dy/dx + a_0(x) y = f(x)

Nonlinear Differential Equations

Include powers greater than one or products of the dependent variable and its derivatives.

Example: (dy/dx)² + y³ = x

Homogeneous Differential Equations

All terms contain the dependent variable or its derivatives.

Example: d²y/dx² + 3 dy/dx + 2y = 0

Non-Homogeneous Differential Equations

Include a non-zero term independent of the dependent variable on the right-hand side.

Example: d²y/dx² + 3 dy/dx + 2y = sin(x)

FAQs

Does the solver show step-by-step solutions?

Yes, it provides detailed, stepwise explanations for every problem.

Can I upload a photo of an equation?

Yes, the tool can automatically recognize and solve equations from images.

Which methods are used to solve differential equations?

Common approaches include separation of variables, integrating factors, characteristic equations, Laplace transforms, and numerical methods.

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