Second-Order ODE Solver Notes

Published on: September 29, 2026

This Second Order ODE Solver Calculator solves a linear equation with constant coefficients that involves the second derivative, and shows the working line by line. Trying a solution of exponential shape turns the equation into an ordinary quadratic, called the characteristic equation. The kind of roots that quadratic has decides the shape of the answer, whether exponential, repeated or oscillating. Following the same pattern every time keeps the three cases apart.

Step-by-step method

  1. Set up the problem.
  2. Form the characteristic equation.
  3. Solve for the roots.
  4. Build the general solution with C1 and C2.

Formula:

\(a y'' + b y' + c y = 0 \;\Rightarrow\; a r^{2} + b r + c = 0\)
See Example 1 Hide Example 1

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  2. Form the characteristic equation.
  3. Solve for the roots.
  4. Build the general solution with C1 and C2.

Step 1

Step 1 - Set up the problem.

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:

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

Final Answer

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

Problem

\(5 y{\left(x \right)} + 2 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Form the characteristic equation.
  3. Solve for the roots.
  4. Build the general solution with C1 and C2.

Step 1

Step 1 - Set up the problem.

In this problem:

\(5 y{\left(x \right)} + 2 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

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:

\(r^{2} + 2 r + 5 = 0\)

Step 4

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

In this problem:

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

Final Answer

\(y = \left(C_{1} \sin{\left(2 x \right)} + C_{2} \cos{\left(2 x \right)}\right) e^{- x}\)
See Example 3 Hide Example 3

Problem

\(6 y{\left(x \right)} - 5 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Form the characteristic equation.
  3. Solve for the roots.
  4. Build the general solution with C1 and C2.

Step 1

Step 1 - Set up the problem.

In this problem:

\(6 y{\left(x \right)} - 5 \frac{d}{d x} y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

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:

\(r^{2} - 5 r + 6 = 0\)

Step 4

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

In this problem:

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

Final Answer

\(y = \left(C_{1} + C_{2} e^{x}\right) e^{2 x}\)
See Example 4 Hide Example 4

Problem

\(4 y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

Approach

Step-by-step method

  1. Set up the problem.
  2. Form the characteristic equation.
  3. Solve for the roots.
  4. Build the general solution with C1 and C2.

Step 1

Step 1 - Set up the problem.

In this problem:

\(4 y{\left(x \right)} + \frac{d^{2}}{d x^{2}} y{\left(x \right)} = 0\)

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:

\(r^{2} + 4 = 0\)

Step 4

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

In this problem:

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

Final Answer

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