Final answer:
To determine if the equation x²y³ + x (1+y²)y' = 0 is exact, we calculate the partial derivatives of M(x,y) and N(x,y), and then apply the integrating factor μ(x,y) = 1/(xy³) to make it exact. After confirming the equation is exact, we find a potential function Φ(x,y) and solve the equation.
Step-by-step explanation:
The student is tasked with showing that the equation x²y³ + x (1+y²)y' = 0 is not exact, but becomes exact when multiplied by an integrating factor μ(x,y) = 1/(xy³). To determine whether the given equation is exact, we first identify the functions M(x,y) = x²y³ and N(x,y) = x(1+y²) which represent the coefficients of dx and dy, respectively. We then calculate the partial derivatives ∂M/∂y and ∂N/∂x and compare them. If the equation is exact, these partial derivatives should be equal.
To make the equation exact, we multiply by the integrating factor μ(x,y) = 1/(xy³), which yields the new M and N functions. We again calculate the partial derivatives ∂M/∂y and ∂N/∂x for these new functions. If the partial derivatives are equal after multiplication, the equation has become exact. The final step is to solve the resulting exact equation, which can be done by finding a potential function Φ(x,y) such that ∂Φ/∂x = M and ∂Φ/∂y = N.