Second-Order ODE Solver Calculator

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

This Second-Order ODE Solver solves linear second-order equations such as y'' - 3y' + 2y = 0. For constant coefficients it forms the characteristic equation, finds the roots, and builds the general solution with C1 and C2.

Step-by-step method

  1. Write the differential equation clearly.
  2. Form the characteristic equation.
  3. Solve for the roots.
  4. Build the general solution with C1 and C2.

Formula:

\(a y'' + b y' + c y = 0 \;\Rightarrow\; a r^{2} + b r + c = 0\)

Example 1:

\\(2 y{\left(x \right)} - 3 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\\)

Step 1 - Write the differential equation clearly.

In this problem: We are solving \(2 y{\left(x \right)} - 3 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\).

\(2 y{\left(x \right)} - 3 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

Step 2 - Form the characteristic equation.

In this problem: For constant coefficients, replace y'' , y' , y with r^2, r, 1.

\(
\(a y'' + b y' + c y = 0 \;\Rightarrow\; a r^{2} + b r + c = 0\)
\)

Step 3 - Solve for the roots.

In this problem: The characteristic equation is \(r^{2} - 3 r + 2 = 0\), with roots \(r = 2,\; r = 1\).

\(r^{2} - 3 r + 2 = 0\)

Step 4 - Build the general solution with C1 and C2.

In this problem: Build the general solution from the roots.

\(y = \left(C_{1} + C_{2} e^{x}\right) e^{x}\)

Final answer:

\\(y = \left(C_{1} + C_{2} e^{x}\right) e^{x}\\)

Example 2:

\\(y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\\)

Step 1 - Write the differential equation clearly.

In this problem: We are solving \(y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\).

\(y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

Step 2 - Form the characteristic equation.

In this problem: For constant coefficients, replace y'' , y' , y with r^2, r, 1.

\(
\(a y'' + b y' + c y = 0 \;\Rightarrow\; a r^{2} + b r + c = 0\)
\)

Step 3 - Solve for the roots.

In this problem: The characteristic equation is \(r^{2} + 1 = 0\), with roots \(r = - i,\; r = i\).

\(r^{2} + 1 = 0\)

Step 4 - Build the general solution with C1 and C2.

In this problem: Build the general solution from the roots.

\(y = C_{1} \sin{\left(x \right)} + C_{2} \cos{\left(x \right)}\)

Final answer:

\\(y = C_{1} \sin{\left(x \right)} + C_{2} \cos{\left(x \right)}\\)
See Example 1 Hide Example 1

Problem

\(2 y{\left(x \right)} - 3 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

Approach

Step-by-step method

  1. Write the differential equation clearly.
  2. Form the characteristic equation.
  3. Solve for the roots.
  4. Build the general solution with C1 and C2.

Step 1

Step 1 - Write the differential equation clearly.

In this problem:

\(2 y{\left(x \right)} - 3 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

Step 2

Step 2 - Form the characteristic equation.

In this problem:

\(
\(a y'' + b y' + c y = 0 \;\Rightarrow\; a r^{2} + b r + c = 0\)
\)

Step 3

Step 3 - Solve for the roots.

In this problem:

\(r^{2} - 3 r + 2 = 0\)

Step 4

Step 4 - Build the general solution with C1 and C2.

In this problem:

\(y = \left(C_{1} + C_{2} e^{x}\right) e^{x}\)

Final Answer

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

Problem

\(y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

Approach

Step-by-step method

  1. Write the differential equation clearly.
  2. Form the characteristic equation.
  3. Solve for the roots.
  4. Build the general solution with C1 and C2.

Step 1

Step 1 - Write the differential equation clearly.

In this problem:

\(y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

Step 2

Step 2 - Form the characteristic equation.

In this problem:

\(
\(a y'' + b y' + c y = 0 \;\Rightarrow\; a r^{2} + b r + c = 0\)
\)

Step 3

Step 3 - Solve for the roots.

In this problem:

\(r^{2} + 1 = 0\)

Step 4

Step 4 - Build the general solution with C1 and C2.

In this problem:

\(y = C_{1} \sin{\left(x \right)} + C_{2} \cos{\left(x \right)}\)

Final Answer

\(y = C_{1} \sin{\left(x \right)} + C_{2} \cos{\left(x \right)}\)