Differential Equation Solver

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Solve first-order linear and separable differential equations.

Equation Format

Model Equation

y'(x) + 2y = 4

Parameters (P and Q)

Coefficient P (in Py)

Forcing Term Q

Initial Condition y(x₀) = y₀

x₀

y₀

Solution Steps

y(x) = 2 - 1.000·exp(-2x)

Particular Solution

General Solutiony(x) = 2 + C·exp(-2x)

Constant (C)-1.0000

Step-by-Step Logic

1.Calculate Integrating Factor: μ(x) = exp(∫P dx) = e^(2x).
2.Multiply both sides by μ(x) and integrate.
3.Apply the point (0, 1) to find C = -1.00.

About this calculator

What are Differential Equations?

A differential equation (DE) is an equation that relates one or more functions and their derivatives. In applications, the functions usually represent physical quantities, the derivatives represent their rates of change, and the DE defines a relationship between the two. Solving a DE means finding the functions that satisfy this relationship.

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Pro Tips

Separable equations can be rearranged to bring all y's to one side and x's to the other.
Linear first-order equations are often solved using an 'Integrating Factor', μ(x) = exp(∫P(x)dx).
The initial condition y(x₀)=y₀ allows you to solve for the constant 'C' and find a unique solution.

Fun Facts

"DEs are the universal language for modeling everything from planetary motion to the spread of diseases."
"Many DEs cannot be solved in terms of elementary functions and require numerical methods like Runge-Kutta."
"The discovery of the laws of motion by Newton and Leibniz essentially created the field of differential equations."

Formula: dy/dx + P(x)y = Q(x)