Cauchy-Euler ODE Solver Calculator

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

This Cauchy-Euler ODE Solver Calculator solves an equation whose coefficients are powers of x matching the order of each derivative, and shows the working line by line. Trying a solution of the form x to the power r turns the equation into an ordinary quadratic in r, called the indicial equation. The roots of that quadratic decide whether the answer uses plain powers, a repeated power with a logarithm, or oscillating terms. Following the same pattern every time keeps the three cases apart.

Step-by-step method

  1. Set up the problem.
  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:

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

Problem

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

Approach

Step-by-step method

  1. Set up the problem.
  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 - Set up the problem.

In this problem:

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

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:

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

Step 4

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

In this problem:

\(y = \frac{C_{1} + C_{2} x^{5}}{x^{3}}\)

Final Answer

\(y = \frac{C_{1} + C_{2} x^{5}}{x^{3}}\)
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. Set up the problem.
  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 - Set up the problem.

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:

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

Step 4

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

In this problem:

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

Final Answer

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

Example Problems

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