Final answer:
To solve the given exact differential equation, we first check if it is exact. Then, we integrate the equation to find the solution function. The solution is given by the equation: x(y^2x) + y^3(x) + y^2(y) + xy^2 + (3/2)(y^2x^2) = C, where C is the constant of integration.
Step-by-step explanation:
To find the solution to the given exact differential equation, we need to check if it is exact and find the integrating factor if necessary. Let's check if it is exact:
Step 1: Calculate the partial derivatives of the function with respect to x (fx) and y (fy):
fx = y^2 + y^3
fy = 3yx^2 + 2xy + y^2
Step 2: Check if the equation fx dx + fy dy = 0 satisfies the condition fy = ∂f/∂y, where f(x, y) is the solution function of the differential equation:
fy = ∂f/∂y = 2xy + y^2
Since fy = ∂f/∂y, the equation is exact. Therefore, we can find the solution function by integrating the equation:
∫(xy^2 + y^3)dx + ∫(3yx^2 + 2xy + y^2)dy = 0
To solve this, integrate the first term with respect to x and the second term with respect to y:
∫xy^2 dx + ∫y^3 dx + ∫3yx^2 dy + ∫2xy dy + ∫y^2 dy = 0
x∫y^2 dx + y^3∫dx + y^2∫dy + 2∫xy dy + 3∫yx^2 dy = 0
x(y^2x) + y^3(x) + y^2(y) + 2(xy^2)/2 + 3(y^2x^2)/2 = C
where C is the constant of integration.