Chapter 1 - First-Order Differential Equations - 1.6 First-Order Linear Differential Equations - Problems - Page 59: 3

$y=-x^5\cos{x}+x^4\sin{x}+c_{1}x^4$

Given that: $x^2\frac{dy}{dx}-4xy=x^7\sin{x}$ Integrating factor $I(x)=e^{\int{-4x^{-1}dx}}=e^{-4logx}=x^{-4}$ So, the integrating factor=$x^{-4}$ Thus; $y.I(x)=\int q(x)I(x)dx+c_{1}$ $yx^{-4}=\int x^{-4}x^5\sin{x}dx+c_{1}$ $y=-x^5\cos{x}+x^4\sin{x}+c_{1}x^4$