What is Differential Equations?

Use this calculator to compute Differential Equations values with step-by-step solutions.

How do I use this calculator?

Enter the required values; results and step-by-step derivation update automatically.

Where is Differential Equations used?

Calculus, calculus, physics, engineering, and related disciplines.

What formulas does this use?

All standard differential equations formulas are applied and shown step-by-step in the results.

Can I get step-by-step solutions?

Yes. The calculator shows a full step-by-step derivation when results are displayed.

How accurate are the results?

Results use standard floating-point arithmetic (~15 significant digits). Suitable for education and professional use.

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CALCULUSCalculusMathematics Calculator

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Differential Equations Calculator — Solve ODEs Step by Step

Solve first-order ordinary differential equations including separable (dy/dx = y) and linear (dy/dx = x + y) types. Supports initial conditions for particular solutions.

Concept Fundamentals

dy/dx = g(x)h(y)

Separable

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

Linear

μ = e^(∫P dx)

Integrating factor

y = Ce^x

dy/dx = y

Did our AI summary help? Let us know.

dy/dx = y → y = Ce^x (exponential growth/decay) dy/dx = x + y → linear, use integrating factor e^(-x) Separable: ∫dy/h(y) = ∫g(x)dx

Key quantities

dy/dx = g(x)h(y)

Separable

Key relation

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

Linear

Key relation

μ = e^(∫P dx)

Integrating factor

Key relation

y = Ce^x

dy/dx = y

Key relation

Ready to run the numbers?

Why: Differential equations model real-world phenomena: population growth, radioactive decay, electrical circuits, and mechanical systems. First-order ODEs are foundational — separable and linear types have standard solution methods.

How: Separable: separate variables and integrate. Linear: use integrating factor μ = e^(∫P dx), multiply through, integrate. Initial conditions determine the constant C.

dy/dx = y → y = Ce^x (exponential growth/decay)dy/dx = x + y → linear, use integrating factor e^(-x)

Sources:Paul's Online Notes - ODEsOpenStax Calculus Vol 2

Run the calculator when you are ready.

Differential Equations CalculatorSolve separable and linear first-order ODEs

First-Order ODE Solver

Solve separable and linear differential equations. Supports dy/dx = y and dy/dx = x + y with initial conditions.

📝 Examples — Click to Load

Input

Equation

Use initial condition

Initial conditionFormat: y(x) = value

Solution

y = 2e^x - x - 1

Particular: y = 2e^x - x - 1

ode_solver.shSOLVED

$ solve_ode "dy/dx = x + y" --ic "y(0) = 1"

General

y = Ce^x - x - 1

Particular

y = 2e^x - x - 1

Differential Equations

dy/dx = x + y

y = 2e^x - x - 1

numbervibe.com/calculators/mathematics/calculus/differential-equations-calculator

Solution Curve y(x)

Solution Sign Distribution

📐 Calculation Steps

IDENTIFY

Identify equation

dy/dx = x + y (linear first-order)

REWRITE

Rewrite

dy/dx - y = x

ext{Form}: ext{dy}/ ext{dx} + P(x)y = Q(x), P(x) = -1, Q(x) = x

METHOD

Integrating factor

μ(x) = e^(-x)

\text{mu} = e^(∫P ext{dx})

SOLUTION

General solution

y = Ce^x - x - 1

y = -x - 1 + ext{Ce}^x

INITIAL CONDITION

Apply y(0) = 1

C = 2

RESULT

Particular solution

y = 2e^x - x - 1

For educational and informational purposes only. Verify with a qualified professional.

🧮 Fascinating Math Facts

📊

dy/dx = y models exponential growth (C>0) or decay (C<0)

— Calculus

🔬

Integrating factor converts linear ODE to exact derivative form

— Paul's Online Notes

📋 Key Takeaways

• Separable: dy/dx = f(x)g(y) → ∫(1/g(y))dy = ∫f(x)dx
• Linear: dy/dx + P(x)y = Q(x) → use integrating factor μ = e^(∫P dx)
• dy/dx = y → y = Ce^x (exponential growth/decay)
• dy/dx = x + y → linear, solved via integrating factor

💡 Did You Know?

📐ODEs model population growth, radioactive decay, and circuit behaviorSource: OpenStax

∫The integrating factor μ(x) = e^(∫P dx) converts a linear ODE into an exact derivativeSource: Paul's Notes

🔬Newton's law of cooling is a first-order linear ODESource: LibreTexts

📈dy/dx = y describes exponential growth (C > 0) or decay (C < 0)Source: Wolfram

📖 How It Works

Separable (dy/dx = y): Rewrite as dy/y = dx, integrate both sides → ln|y| = x + C → y = Ce^x.

Linear (dy/dx = x + y): Rewrite as dy/dx - y = x. Integrating factor μ = e^(-x). Multiply, integrate → y = Ce^x - x - 1.

🎯 Expert Tips

💡 Identify the type

Separable: variables can be split. Linear: form dy/dx + P(x)y = Q(x).

💡 Initial conditions

Use y(x₀) = y₀ to find C and get a particular solution.

⚖️ Comparison Table

Type	Form	Method
Separable	dy/dx = f(x)g(y)	Separate & integrate
Linear	dy/dx + P(x)y = Q(x)	Integrating factor

❓ Frequently Asked Questions

What is a separable differential equation?

One that can be written as dy/g(y) = f(x)dx. Both sides can then be integrated separately.

What is an integrating factor?

μ(x) = e^(∫P(x)dx). Multiplying the linear ODE by μ converts it to d/dx[μy] = μQ, which is easy to integrate.

Why does dy/dx = y give y = Ce^x?

Separating: dy/y = dx. Integrating: ln|y| = x + C₁. Exponentiating: y = Ce^x where C = ±e^C₁.

📊 Infographic Stats

1st

Order (dy/dx)

Integrating factor

Arbitrary constant

e^x

Exponential solution

📚 Official Sources

Paul's Online Notes - ODEs ↗

Separable and linear ODE tutorials

OpenStax Calculus Vol 2 ↗

First-order linear equations

Mathematics LibreTexts ↗

Introduction to differential equations

Wolfram MathWorld - ODE ↗

Differential equation reference

⚠️ Disclaimer: This calculator supports a limited set of first-order ODEs. For complex equations, use numerical methods or specialized software.

👈 START HERE

⬅️Jump in and explore the concept!

Differential Equations Calculator — Solve ODEs Step by Step

First-Order ODE Solver

📝 Examples — Click to Load

Input

Solution Curve y(x)

Solution Sign Distribution

📐 Calculation Steps

🧮 Fascinating Math Facts

📋 Key Takeaways

💡 Did You Know?

📖 How It Works

🎯 Expert Tips

💡 Identify the type

💡 Initial conditions

⚖️ Comparison Table

❓ Frequently Asked Questions

What is a separable differential equation?

What is an integrating factor?

Why does dy/dx = y give y = Ce^x?

📊 Infographic Stats

📚 Official Sources

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