Question

Exact Solution A) (4x³y³+1/x)dx+(3x⁴y²−Y1)dy=0 For X(E)=1

Answer 1

To solve the given exact equation, we need to check if it meets the exactness condition: ∂(M)/∂(y) = ∂(N)/∂(x). If the condition is met, we can solve for the exact solution by integrating M and finding a function ₁1(x). Then, we differentiate ₁1(x) with respect to y and find its antiderivative yIII(x).

Step-by-step explanation:

The given equation is (4x³y³+1/x)dx+(3x⁴y²−Y1)dy=0

Step 1: To solve the given exact equation, we need to check if it meets the exactness condition: ∂(M)/∂(y) = ∂(N)/∂(x).

Step 2: We differentiate both M and N with respect to y and x respectively. In this case, ∂(M)/∂(y) = 12x³y², and ∂(N)/∂(x) = 12x³y². Since the partial derivatives are equal, the condition for exactness is met.

Step 3: To solve for the exact solution, we integrate M with respect to x, holding y constant, and find a function ₁1(x). Then we differentiate ₁1(x) with respect to y and find its antiderivative yIII(x).