First-Order ODE Solver Calculator

Published on: July 18, 2026
Final Answer: Free • Full Steps: Pro

This First Order ODE Solver Calculator finds the general solution of an equation that links y to its first derivative, and shows the working line by line. The type of equation is identified first, because a separable equation, a linear one and an exact one are each undone by a different move. Integrating then introduces the arbitrary constant that every general solution carries. Following the same pattern every time makes the choice of method visible rather than guessed.

Step-by-step method

  1. Set up the problem.
  2. Identify the first-order type (separable, linear, and so on).
  3. Integrate to get the general solution with C1.
  4. Write the general solution y(x).

Formula:

\(\frac{dy}{dx} + P(x)\, y = Q(x) \;\Rightarrow\; \mu = e^{\int P \, dx}\)
See Example 1 Hide Example 1

Problem

\(\frac{d}{d x} y{\left(x \right)} = x y{\left(x \right)}\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Identify the first-order type (separable, linear, and so on).
  3. Integrate to get the general solution with C1.
  4. Write the general solution y(x).

Step 1

Step 1 - Set up the problem.

In this problem:

\(\frac{d}{d x} y{\left(x \right)} = x y{\left(x \right)}\)

Step 2

Step 2 - Identify the first-order type (separable, linear, and so on).

In this problem:

\(\frac{dy}{dx} + P(x)\, y = Q(x) \;\Rightarrow\; \mu = e^{\int P \, dx}\)

Step 3

Step 3 - Integrate to get the general solution with C1.

In this problem:

\(y = C_{1} e^{\frac{x^{2}}{2}}\)

Step 4

Step 4 - Write the general solution y(x).

In this problem:

\(y = C_{1} e^{\frac{x^{2}}{2}}\)

Final Answer

\(y = C_{1} e^{\frac{x^{2}}{2}}\)
See Example 2 Hide Example 2

Problem

\(y{\left(x \right)} + \frac{d}{d x} y{\left(x \right)} = x\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Identify the first-order type (separable, linear, and so on).
  3. Integrate to get the general solution with C1.
  4. Write the general solution y(x).

Step 1

Step 1 - Set up the problem.

In this problem:

\(y{\left(x \right)} + \frac{d}{d x} y{\left(x \right)} = x\)

Step 2

Step 2 - Identify the first-order type (separable, linear, and so on).

In this problem:

\(\frac{dy}{dx} + P(x)\, y = Q(x) \;\Rightarrow\; \mu = e^{\int P \, dx}\)

Step 3

Step 3 - Integrate to get the general solution with C1.

In this problem:

\(y = C_{1} e^{- x} + x - 1\)

Step 4

Step 4 - Write the general solution y(x).

In this problem:

\(y = C_{1} e^{- x} + x - 1\)

Final Answer

\(y = C_{1} e^{- x} + x - 1\)

Example Problems

Click one to load it into the box and solve it.