Question

In Problems, 1-20 determine whether the given differential equation is exact. If it is exact, solve it.

$\mathbf{(}\mathbf{2}\mathit{y}\mathit{s}\mathit{i}\mathit{n}\mathit{x}\mathit{c}\mathit{o}\mathit{s}\mathit{x}\mathbf{-}\mathit{y}\mathbf{+}\mathbf{2}{\mathit{y}}^{\mathbf{2}}{\mathit{e}}^{\mathbf{x}{\mathbf{y}}^{\mathbf{2}}}\mathbf{)}\mathit{d}\mathit{x}\mathbf{=}\mathbf{(}\mathit{x}\mathbf{-}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{x}\mathbf{-}\mathbf{4}\mathit{x}\mathit{y}{\mathit{e}}^{\mathbf{x}{\mathbf{y}}^{\mathbf{2}}}\mathbf{)}\mathit{d}\mathit{y}$

Short answer

The answer is$-xy+ysi{n}^{2}x+2e^{x{y}^2}=c$

Step-by-step solution

Given Information

The given equation is, $(2ysinxcosx-y+2y^2{e}^{x{y}^2})dx=(x-si{n}^{2}x-4xy{e}^{x{y}^2})dy$.

Which can also be written as .

$\left((2ysinxcosx-y+2y^2{e}^{x{y}^2}\right)dx-\left(x+si{n}^{2}x+4xy{e}^{x{y}^2}\right)dy=0$

Compare the given equation with $Mdx+Ndy=0$.

$$\begin{gathered} M(x,y)=2ysinxcosx-y+2y^2+e^{x{y}^2} \\ N(x,y)=-x+si{n}^{{}^2}x+4xy{e}^{x{y}^2} \end{gathered}$$

Determining the exactness of the differential equation

Checking whether the differential equation is exact by finding $\frac{\partial M}{\partial y},\frac{\partial N}{\partial x}$

$\frac{\partial M}{\partial y}=2sinxcosx-1+4y{e}^{x{y}^2}+4x{y}^3{e}^{x{y}^2}$

$\frac{\partial N}{\partial x}=-1+2sinxcosx+4y{e}^{x{y}^2}+4x{y}^3{e}^{x{y}^2}$

As a result, the equation is exact $\frac{\partial M}{\partial y}=\frac{\partial N}{\partial x}$.

Next, we have to find the solution of the given equation. Where$f(x,y)=-xy+ysi{n}^{2}x+2e^{x{y}^2}+g\left(x\right)$

Integrate

Now, consider the equation .$f(x,y)=-xy+ysi{n}^{2}x+2e^{x{y}^2}+g\left(x\right)$

$f(x,y)=-xy+ysi{n}^{2}x+2e^{x{y}^2}+g\left(x\right)$

Take $\frac{\partial f}{\partial x}$, then

$\frac{\partial f}{\partial x}=-y+2ysinxcosx+2y^{{}^2}+e^{x{y}^2}+g\text{'}\left(x\right)$

Since then, $M(x,y)=\frac{\partial f}{\partial x}$, We have notice that $g\text{'}\left(x\right)=0$

$\begin{array}{c}g\text{'}\left(x\right)=0\\ \Rightarrow g\left(x\right)=c\end{array}$

.

So, the result is $-xy+ysi{n}^{2}x+2e^{x{y}^2}=c$.