solve 2 dy/dx + y tan x = (4x + 5)^2 / ( cos x) x y^3

Question

solve calculas first order differetiation either can be seperable ,,, or linear ,, or exact ,,,or homogeneous ,,,or bernoullis equation

Answer 1

Multiply through by y^3cos(x): 2y^3cos(x)dy/dx+y^4sin(x)=(4x+5)^2/x=16x+40+25/x.

Let p=y^4cos(x), then dp/dx=4y^3cos(x)dy/dx-y^4sin(x);

2y^3cos(x)dy/dx=(dp/dx+y^4sin(x))/2=(dp/dx+ptan(x))/2.

dp/dx+3ptan(x)=32x+80+50/x; cos(x)dp/dx+3psin(x)=32xcos(x)+80cos(x)+50cos(x)/x.

u(x)dp/dx+3u(x)ptan(x)=u(x)(32x+80+50/x), where u(x) is a function of x to be determined.

d(up)/dx=udp/dx+pdu/dx; so pdu/dx=3uptan(x); du/dx=3utan(x); ∫du/u=∫3tan(x)dx; ln(u)=-3ln(cos(x)); u=1/cos^3(x) and up=y^4sec^2(x)=∫(32x+80+50/x)dx/cos^3(x).

More to follow...