# Legendre Linear Equation

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**Legendre Linear Equation:**

**It is the Form:**

Where A_{1},A_{2},..........,A_{n} are constants. It can be reduced to linear differential equation with constant coefficients.

**By taking:**

**Consider: **

Substitue (2) in (1) gives: the linear differential equation of constant coefficients.

**Examples: **Solve (2x 3)^{2}y'' - (2x 3)y' - 12y = 6x

**Solution:** This is Legendre's linear equation:

(2x 3)^{2}y'' - (2x 3)y' - 12y = 6x................(1)

Put z = log (2x 3), e^{z} = 2x 3

(2x 3)D = 2θ

**Solution is:**