Verify that the given differential equation is not exact....

Verify that the given differential equation is not exact. Multiply the given differential equation by the indicated integrating factor $\mu(x, y)$ and verify that the new equation is exact. Solve. $$ \left(x^{2}+2 x y-y^{2}\right) d x+\left(y^{2}+2 x y-x^{2}\right) d y=0 ; \mu(x, y)=(x+y)^{-2} $$

Verify that the given differential equation is not exact. Multiply the given differential equation by the indicated integrating factor $\mu(x, y)$ and verify that the new equation is exact. Solve.
$$
\left(x^{2}+2 x y-y^{2}\right) d x+\left(y^{2}+2 x y-x^{2}\right) d y=0 ; \mu(x, y)=(x+y)^{-2}
$$

Key Concepts

Potential Functions and Integration

Once a differential equation is made exact (either originally or by applying an integrating factor), the next step is to find a potential function F(x, y) such that its total differential equals the left-hand side of the equation. The method generally involves integrating one of the functions M(x, y) or N(x, y) with respect to its variable and then determining the complementary function by differentiating and comparing with the other term, ultimately yielding the solution in an implicit form.

Integrating Factors

An integrating factor is a function, usually of x, y, or both, that when multiplied by a non?exact differential equation transforms it into an exact one. This method is crucial for solving a wider variety of first-order differential equations, and the integrating factor is often found by considering criteria dictated by the structure of the original equation.

Exact Differential Equations

This concept involves differential equations expressed in the form M(x, y) dx + N(x, y) dy = 0 where there exists a function F(x, y) such that the total differential dF equals M dx + N dy. The equation is called 'exact' if the mixed partial derivatives F_xy and F_yx are equal, which implies that the equation can be integrated directly to find a potential function yielding the solution.

Verification of Exactness

In the context of differential equations, verifying exactness means checking whether the condition ?M/?y = ?N/?x is satisfied. If this equality holds, the equation is exact and directly integrable; if not, additional techniques like the use of an integrating factor are necessary to convert the equation into an exact form.

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