Question

Determine whether the given differential equation is exact. If it is exact, solve it. $$\begin{aligned} &\left(2 y \sin x \cos x-y+2 y^{2} e^{x y^{2}}\right) d x\\ &=\left(x-\sin ^{2} x-4 x y e^{x y^{2}}\right) d y \end{aligned}$$

          Determine whether the given differential equation is exact. If it is exact, solve it.
$$\begin{aligned}
&\left(2 y \sin x \cos x-y+2 y^{2} e^{x y^{2}}\right) d x\\
&=\left(x-\sin ^{2} x-4 x y e^{x y^{2}}\right) d y
\end{aligned}$$

Added by Kristin R.

Differential Equations and Linear Algebra

Stephen W. Goode, Scott A. Annin 4th Edition

Instant Answer

Solved by Expert Madhur L

07/28/2023

Step 1

To solve the equation, we can try to find an integrating factor $\mu(x,y)$ that makes the equation exact. We can use the formula: $$\mu(x,y)=\frac{1}{M}\left(\frac{\partial N}{\partial x}-\frac{\partial M}{\partial y}\right)$$ where $M$ and $N$ are the Show more…

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Determine whether the given differential equation is exact. If it is exact, solve it. $$egin{aligned} &left(2 y sin x cos x-y+2 y^{2} e^{x y^{2}} ight) d x\ &=left(x-sin ^{2} x-4 x y e^{x y^{2}} ight) d y end{aligned}$$