Calculators / Differential Equations / Cauchy–Euler ODE Solver

Cauchy–Euler ODE Solver

Published on: January 4, 2026

This Cauchy–Euler ODE Solver helps you solve Cauchy–Euler differential equations and shows each step clearly. It works by recognising the variable-coefficient ODE form, using the standard trial solution method, and optionally applying initial conditions to find a particular solution. This makes it useful for checking answers, understanding how the Cauchy–Euler method works, and practising differential equations step by step.

Step-by-step method

  1. Rewrite into Euler–Cauchy form: a x^2 y'' + b x y' + c y = f(x).
  2. Divide by a to get: x^2 y'' + α x y' + β y = r(x).
  3. If homogeneous, try y = x^m and build the characteristic equation.
  4. Solve for m and write the general solution based on root type.
  5. If initial conditions are given, solve for constants.

Formula bank:

Cauchy–Euler: x²y″ + αxy′ + βy = r(x)
Trial: y = x^m
y′ = m x^(m−1), y″ = m(m−1) x^(m−2)
Characteristic: m(m−1) + αm + β = 0
m1≠m2: y = C1 x^m1 + C2 x^m2
m repeated: y = (C1 + C2 ln x) x^m
m=r±iω: y = x^r (C1 cos(ω ln x) + C2 sin(ω ln x))

Example 1: x^2*y'' - 3*x*y' + 4*y = 0, x, y

Step 1 - Write the ODE.

In this problem: Start with the given differential equation.

4y + x2
d2y
dx2
− 3x
dy
dx
= 0

Step 2 - Confirm Euler–Cauchy form.

In this problem: Match a x²y″ + b x y′ + c y = f(x).

Cauchy–Euler: x²y″ + αxy′ + βy = r(x)
Trial: y = x^m
y′ = m x^(m−1), y″ = m(m−1) x^(m−2)
Characteristic: m(m−1) + αm + β = 0
m1≠m2: y = C1 x^m1 + C2 x^m2
m repeated: y = (C1 + C2 ln x) x^m
m=r±iω: y = x^r (C1 cos(ω ln x) + C2 sin(ω ln x))

Step 3 - Divide into standard form.

In this problem: x²y″ + αxy′ + βy = r(x).

a = 1, b = -3, c = 4
α = -3, β = 4
r(x) = 0

Step 4 - Use y = x^m.

In this problem: Build the characteristic equation.

4 + m2 − 4m = 0

Step 5 - Solve for m.

In this problem: Find the roots of the characteristic equation.

m = 2

Step 6 - Write the general solution.

In this problem: Use the root type to choose the correct form.

y(x) = x2(C1 + C2ln(x))

Step 7 - Final answer.

In this problem: Write the solution.

y(x) = x2(C1 + C2ln(x))

Final answer: See final step

Example 2: x^2*y'' + x*y' - y = 0, x, y, y(1)=2, y'(1)=0

Step 1 - Write the ODE.

In this problem: Start with the given differential equation.

−y + x
dy
dx
+ x2
d2y
dx2
= 0

Step 2 - Confirm Euler–Cauchy form.

In this problem: Match a x²y″ + b x y′ + c y = f(x).

Cauchy–Euler: x²y″ + αxy′ + βy = r(x)
Trial: y = x^m
y′ = m x^(m−1), y″ = m(m−1) x^(m−2)
Characteristic: m(m−1) + αm + β = 0
m1≠m2: y = C1 x^m1 + C2 x^m2
m repeated: y = (C1 + C2 ln x) x^m
m=r±iω: y = x^r (C1 cos(ω ln x) + C2 sin(ω ln x))

Step 3 - Divide into standard form.

In this problem: x²y″ + αxy′ + βy = r(x).

a = 1, b = 1, c = -1
α = 1, β = -1
r(x) = 0

Step 4 - Use y = x^m.

In this problem: Build the characteristic equation.

−1 + m2 = 0

Step 5 - Solve for m.

In this problem: Find the roots of the characteristic equation.

m = -1, 1

Step 6 - Write the general solution.

In this problem: Use the root type to choose the correct form.

y(x) = C1
1
x
+ C2x

Step 7 - Apply initial conditions.

In this problem: Solve for C1 and C2.

y(1) = 2
y′(1) = 0

Step 8 - Final answer.

In this problem: Write the solution.

y(x) = x +
1
x

Final answer: See final step