Which of the following first-order differential equations are exact? Select all that apply.
A) $ (8 y-4 x) y^{\prime}-4 y=0 $
B) $ (2 y-5 x)+(5 y+2 x) y^{\prime}=0 $
C) $ 4 y^{3}+\left(x^{4}+y^{4}-7 x\right) y^{\prime}=0 $
D) $ \left(7 x-\cos \left(\frac{5}{2} y\right)\right)-\sin \left(\frac{5}{2} y\right) y^{\prime}=0 $

Question

Quizplus · Accepted Answer

Explanation:An equation $ M(x,y)+N(x,y)y^{\prime}=0 $ is exact if $ \frac{\partial M}{\partial y}=\frac{\partial N}{\partial x} $.For A, $ \frac{\partial M}{\partial y}=8 $ and $ \frac{\partial N}{\partial x}=-4 $, so the equation is exact.For B, $ \frac{\partial M}{\partial y}=2 $ and $ \frac{\partial N}{\partial x}=5 $, so the equation is exact.For C, $ \frac{\partial M}{\partial y}=12 y^{2} $ and $ \frac{\partial N}{\partial x}=4 x^{3} $, so the equation is not exact.For D, $ \frac{\partial M}{\partial y}=-\frac{5}{2} \sin \left(\frac{5}{2} y\right) $ and $ \frac{\partial N}{\partial x}=0 $, so the equation is not exact.

Which of the Following First-Order Differential Equations Are Exact? Select (8y−4x)y′−4y=0 (8 y-4 x) y^{\prime}-4 y=0 (8y−4x)y′−4y=0

Which of the Following First-Order Differential Equations Are Exact? Select $(8 y-4 x) y^{\prime}-4 y=0$