Calculus

(2x + y) dx + (x + 2y) dy = 0

Step-by-step solution with explanation

Final Answer

, where C is an arbitrary constant.

Step-by-step solution

Check if the equation is exact

We identify M and N from the form M dx + N dy = 0. To be exact, we need the partial derivative of M with respect to y to equal the partial derivative of N with respect to x.

Compute partial derivatives to confirm exactness

Since both partial derivatives equal 1, we have an exact equation. This means there exists a potential function F(x, y) such that dF = 0, meaning F = C.

Integrate M with respect to x

We integrate M with respect to x, treating y as a constant. The unknown function g(y) plays the role of the 'constant' of integration, since it can depend on y.

Differentiate F with respect to y and set equal to N

We take the partial derivative of F with respect to y and set it equal to N. This lets us solve for g'(y).

Solve for g(y)

Subtracting x from both sides gives g'(y) = 2y. Integrating with respect to y gives g(y) = y². We absorb the constant into the final constant C.

Write the general solution

We plug g(y) back into F(x, y). Since dF = 0 along solutions, F must equal a constant C. This is the implicit general solution to the differential equation.

Understanding this problem

Learning Insight

An exact equation is one where M dx + N dy is a perfect differential of some function F. The condition ∂M/∂y = ∂N/∂x guarantees this. Solving it is like 'reverse-engineering' a multivariable function whose total differential gives you exactly what you started with.

Quick Tip

Once you confirm exactness, integrate M in x, then differentiate your result in y and match it to N — that two-step move always finds g(y) quickly.

Common Mistake

Students often forget that the 'constant' after integrating M in x can be a full function of y, not just a number. Writing g(y) instead of just C is essential, or you'll miss terms when matching N.