Which of the following is an exact differential equation?

A) $( x + y ) d x + \left( 2 y + x ^ { 2 } \right) d y$
B) $( x + y ) d x + ( 2 y + x ) d y$
C) $\left( x ^ { 2 } + 2 x y \right) d x + ( 2 y + x ) d y$
D) $( x + y ) d x + \left( 2 y ^ { 2 } + 3 x y \right) d y$
E) $\left( x + y ^ { 2 } \right) d x + ( 2 y + x ) d y$

Question

Which of the following is an exact differential equation?

A) $( x + y ) d x + \left( 2 y + x ^ { 2 } \right) d y$
B) $( x + y ) d x + ( 2 y + x ) d y$
C) $\left( x ^ { 2 } + 2 x y \right) d x + ( 2 y + x ) d y$
D) $( x + y ) d x + \left( 2 y ^ { 2 } + 3 x y \right) d y$
E) $\left( x + y ^ { 2 } \right) d x + ( 2 y + x ) d y$

Quizplus · Accepted Answer

For a differential equation of the form $M(x, y)dx + N(x, y)dy = 0$ to be exact, the partial derivative of $M$ with respect to $y$ must equal the partial derivative of $N$ with respect to $x$. In option B, $M(x, y) = x + y$ and $N(x, y) = 2y + x$, both of which have partial derivatives of 1 with respect to $y$ and $x$, respectively, making it an exact differential equation.

Which of the Following Is an Exact Differential Equation? (x+y)dx+(2y+x2)dy( x + y ) d x + \left( 2 y + x ^ { 2 } \right) d y(x+y)dx+(2y+x2)dy

Which of the Following Is an Exact Differential Equation?
$( x + y ) d x + \left( 2 y + x ^ { 2 } \right) d y$