Question

What is the particular solution to the differential equation dy/dx =3 cos(x)y with the initial condition y(pi/2)=-2?

 

Answer 1

dy/dx=3cos(x)y,

dy/y=3cos(x)dx,

ln(y)=3sin(x)+C, where C is a constant,

y=e^(3sin(x)+C) which is always positive, so y cannot be -2.

Perhaps you meant dy/dx=3cos(x)/y?

If so:

ydy=3cos(x)dx,

Integrating:

y²/2=3sin(x)+C.

Plug in x=π/2 and y=-2:

2=3+C, C=-1, y²/2=3sin(x)-1.

This can be written: y²=6sin(x)-2.