Differential Equation

Question

Discuss the exactness of the differential model (2x * cos y + 3x ^ 2 * y) * dx + (x ^ 3 - x ^ 2 * sin y - y) * dy = 0, f find its solution

Answer 1

(2xcosy + 3x²y)dx + (x³ - x²siny - y)dy = 0;

∂/∂y(2xcosy + 3x²y) = (-2xsiny + 3x²) = ∂/∂x(x³ - x²siny - y). So, we have exact equation.

F(x,y) = ∫(2xcosy + 3x²y)dx = x²cosy + x³y + f(y);

dF/dy = - x²siny + x³ + f'(y) = x³ - 2xsiny - y; f'(y) = - y; f(y) = - y²/2 + C

F(x,y) = x²cosy + x³y - y²/2 + C