Let \(y^4= z\) then, \(4y^3 y'= dz/dx\)

\(y^3y'+1/x y^4= sinx/x^4 => 1/4 dz/dx+1/x z=sinx/x^4\)

\(dz/dx+4/x z= (4sinx)/x^4\)

The given ordinary equation is of the form

\(dz/dx+P(x)z=Q(x)\)

Integrating factor: \(IF= e^(int Pdx)= e(int 4/x)dx= x^4\)

\(d/dx(IFz)=IFQ(x)\)

\(IF*z= int IF*1(x)dx\)

\(x^4*z= int(x^4)((4sinx)/x)dx\)

Multiply the original differential equation by the integral factor

\(d/dx(IF*z)= IF*Q(x)\)

\(IF*z= int IF*Q(x)dx\)

\(x^4*z=int (x^4)((4sinx)/x^4)dx\)

\(x^4*z= int (4sinx)dx\)

\(x^4z= 4 int sin xdx\)

\(x^4z= -4cosx+c\)

\(z= (-4cosx)/x^4+c/x^4\)

Put the value of z in the obtained equation

\(z= (-4cosx)/x^4+c/x^4\)

\(z= y^4\)

\(y^4= (-4cosx)/x^4+c/x^4\)

\(y= root(1/4)(((-4cosx)/x^4+c/x^4))\)

\(y^3y'+1/x y^4= sinx/x^4 => 1/4 dz/dx+1/x z=sinx/x^4\)

\(dz/dx+4/x z= (4sinx)/x^4\)

The given ordinary equation is of the form

\(dz/dx+P(x)z=Q(x)\)

Integrating factor: \(IF= e^(int Pdx)= e(int 4/x)dx= x^4\)

\(d/dx(IFz)=IFQ(x)\)

\(IF*z= int IF*1(x)dx\)

\(x^4*z= int(x^4)((4sinx)/x)dx\)

Multiply the original differential equation by the integral factor

\(d/dx(IF*z)= IF*Q(x)\)

\(IF*z= int IF*Q(x)dx\)

\(x^4*z=int (x^4)((4sinx)/x^4)dx\)

\(x^4*z= int (4sinx)dx\)

\(x^4z= 4 int sin xdx\)

\(x^4z= -4cosx+c\)

\(z= (-4cosx)/x^4+c/x^4\)

Put the value of z in the obtained equation

\(z= (-4cosx)/x^4+c/x^4\)

\(z= y^4\)

\(y^4= (-4cosx)/x^4+c/x^4\)

\(y= root(1/4)(((-4cosx)/x^4+c/x^4))\)