solve equation for xsin^2 x=du/dx-cos^2x*dv/dx u=2x^3+3x^2 v=x^2

To solve the equation, we'll have to differentiate u and
v, with respect to x.

du/dx =
d(2x^3+3x^2)/dx

du/dx = 6x^2 +
6x

dv/dx = d(x^2)/dx

dv/dx =
2x

We'll substitute du/dx and dv/dx by their
expression.

(sin x)^2 = 6x^2 + 6x - 2x*(cos
x)^2

We'll move 2x*(cos x)^2 to the left
side.

(sin x)^2 + 2x*(cos x)^2 =
6x(x+1)

But, (cos x)^2 = 1 - (sin
x)^2

(sin x)^2 + 2x - 2x(sin x)^2 =
6x(x+1)

(sin x)^2(1 - 2x) =
2x(3x+3-1)

(sin x)^2(1 - 2x) =
2x(3x+2)

(sin x)^2 = 2x(3x+2)/(1 -
2x)

sin x = 0 for x =
0

2x(3x+2)/(1 - 2x)= 0 for x =
0

The common solution of the given equation
is x = 0.