Cauchy-Euler ODE Solver Calculator

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

This Cauchy-Euler ODE Solver solves equidimensional equations of the form a x^2 y'' + b x y' + c y = 0. It substitutes y = x^r to form the indicial equation, solves for the roots, and builds the general solution.

Step-by-step method

  1. Write the Cauchy-Euler equation clearly.
  2. Substitute y = x^r to form the indicial equation.
  3. Solve for the roots.
  4. Build the general solution with C1 and C2.

Formula:

\(a x^{2} y'' + b x y' + c y = 0 \;\Rightarrow\; a r(r-1) + b r + c = 0\)

Example 1:

\\(x^{2} \frac{d^{2}}{d x^{2}} y{\left(x \right)} - 3 x \frac{d}{d x} y{\left(x \right)} + 4 y{\left(x \right)} = 0\\)

Step 1 - Write the Cauchy-Euler equation clearly.

In this problem: We are solving the Cauchy-Euler equation \(x^{2} \frac{d^{2}}{d x^{2}} y{\left(x \right)} - 3 x \frac{d}{d x} y{\left(x \right)} + 4 y{\left(x \right)} = 0\).

\(x^{2} \frac{d^{2}}{d x^{2}} y{\left(x \right)} - 3 x \frac{d}{d x} y{\left(x \right)} + 4 y{\left(x \right)} = 0\)

Step 2 - Substitute y = x^r to form the indicial equation.

In this problem: Substitute y = x^r; the equation reduces to the indicial equation.

\(
\(a x^{2} y'' + b x y' + c y = 0 \;\Rightarrow\; a r(r-1) + b r + c = 0\)
\)

Step 3 - Solve for the roots.

In this problem: The indicial equation is \(r^{2} - 4 r + 4 = 0\), with roots \(r = 2\).

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

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

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

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

Final answer:

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

Example 2:

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

Step 1 - Write the Cauchy-Euler equation clearly.

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

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

Step 2 - Substitute y = x^r to form the indicial equation.

In this problem: Substitute y = x^r; the equation reduces to the indicial equation.

\(
\(a x^{2} y'' + b x y' + c y = 0 \;\Rightarrow\; a r(r-1) + b r + c = 0\)
\)

Step 3 - Solve for the roots.

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

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

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

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

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

Final answer:

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

Problem

\(x^{2} \frac{d^{2}}{d x^{2}} y{\left(x \right)} - 3 x \frac{d}{d x} y{\left(x \right)} + 4 y{\left(x \right)} = 0\)

Approach

Step-by-step method

  1. Write the Cauchy-Euler equation clearly.
  2. Substitute y = x^r to form the indicial equation.
  3. Solve for the roots.
  4. Build the general solution with C1 and C2.

Step 1

Step 1 - Write the Cauchy-Euler equation clearly.

In this problem:

\(x^{2} \frac{d^{2}}{d x^{2}} y{\left(x \right)} - 3 x \frac{d}{d x} y{\left(x \right)} + 4 y{\left(x \right)} = 0\)

Step 2

Step 2 - Substitute y = x^r to form the indicial equation.

In this problem:

\(
\(a x^{2} y'' + b x y' + c y = 0 \;\Rightarrow\; a r(r-1) + b r + c = 0\)
\)

Step 3

Step 3 - Solve for the roots.

In this problem:

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

Step 4

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

In this problem:

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

Final Answer

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

Problem

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

Approach

Step-by-step method

  1. Write the Cauchy-Euler equation clearly.
  2. Substitute y = x^r to form the indicial equation.
  3. Solve for the roots.
  4. Build the general solution with C1 and C2.

Step 1

Step 1 - Write the Cauchy-Euler equation clearly.

In this problem:

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

Step 2

Step 2 - Substitute y = x^r to form the indicial equation.

In this problem:

\(
\(a x^{2} y'' + b x y' + c y = 0 \;\Rightarrow\; a r(r-1) + b r + c = 0\)
\)

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:

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

Final Answer

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