Question

solve the initial value problem x_dx^dy-2y=〖2x〗^4, y(2)=8

Answer 1

$xy'-2y=(2x)^4$

Solution of homogenуous equation we will find in the form $y=Ax^k$ :

$kAx^k-2Ax^k=0$

$k=0$

$y_{h}=Ax^2$

Solution of inhomogeneous equation we will find as $y=B(x)x^2$ :

$y'=B'(x)x^2+2B(x)x$

After substitution into equation:

$B'(x)x^3+2B'(x)x^2-2B(x)x^2=(2x)^4$

$B'(x)=16x$

$B(x)=8x^2+C$

$y=(8x^2+C)x^2$

$y(2)=(8*4+C)*4=8$

$C=2-32=-30$

Answer: $y=8x^4-30x^2$