Consider the differential equation x2 y (x2 2x)y (x 2)y x3 . Verify that y1 x is one

Chapter 3, Problem 32

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Consider the differential equation \(x^{2} y^{\prime \prime}-\left(x^{2}+2 x\right) y^{\prime}+(x+2) y=x^{3}\) . Verify that \(y_{1}=x\) is one solution of the associated homogeneous equation. Then show that the method of reduction of order discussed in Section 3.2 leads both to a second solution \(y_{2}\) of the homogeneous equation and to a particular solution \(y_{p}\) of the nonhomogeneous equation. Form the general solution of the DE on the interval \((0, \infty)\).

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