The general solution of the exact differential equation $( x + y ) d x + x d y = 0$ is

A) $- x ^ { 2 } + 3 x y = C$
B) $2 x ^ { 2 } - x y = C$
C) $2 x ^ { 2 } + x y = C$
D) $x ^ { 2 } + 2 x y = C$
E) $x ^ { 2 } - 2 x y = C$

Question

The general solution of the exact differential equation $( x + y ) d x + x d y = 0$ is

A) $- x ^ { 2 } + 3 x y = C$
B) $2 x ^ { 2 } - x y = C$
C) $2 x ^ { 2 } + x y = C$
D) $x ^ { 2 } + 2 x y = C$
E) $x ^ { 2 } - 2 x y = C$

Quizplus · Accepted Answer

To solve the given exact differential equation $(x + y)dx + xdy = 0$, integrate $(x + y)dx$ with respect to $x$ and $xdy$ with respect to $y$. Integrating $(x + y)dx$ gives $\frac{1}{2}x^2 + xy + C_1$, and integrating $xdy$ gives $xy + C_2$. Combining these and ignoring the constants $C_1$ and $C_2$ since they can be combined into a single constant $C$, we get $x^2 + 2xy = C$, which matches option D.

The General Solution of the Exact Differential Equation (x+y)dx+xdy=0( x + y ) d x + x d y = 0(x+y)dx+xdy=0

The General Solution of the Exact Differential Equation $( x + y ) d x + x d y = 0$