# Worksheet: Cauchy–Euler Differential Equation

In this worksheet, we will practice solving Cauchy–Euler differential equations of the general form aₙ xⁿ y⁽ⁿ⁾ + aₙ₋₁ xⁿ⁻¹ y⁽ⁿ⁻¹⁾ + ... + a₀ y = f(x).

Q1:

The functions , , and are three linearly independent solutions of the differential equation . Find a particular solution satisfying the initial conditions , , and .

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Q2:

The functions , , and are three linearly independent solutions of the differential equation . Find a particular solution satisfying the initial conditions , , and .

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Q3:

Find the general solution for the following ordinary differential equation with variable coefficients: . This is an example of an Euler-Cauchy differential equation (nonzero right-hand side).

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Q4:

Find the general solution for the homogeneous ordinary differential equation with variable coefficients . This is an example of an Euler–Cauchy differential equation (zero right-hand side).

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Q5:

Find the general solution for the following ordinary differential equation using the reduction of order method: , and .

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