MAT244--2018F > Final Exam

FE-P1

**Victor Ivrii**:

Typed solutions only. No uploads

Find the general solution

\begin{equation*}

\bigl[2x\sin(y) +1\bigr]\,dx +

\bigl[4x^2\cos(y) + 3x\cot(y)+5 \sin(2y)\bigr]\,dy=0\,.

\end{equation*}

Hint: Use the integrating factor.

**Jingze Wang**:

Let $M=2x\sin(y)+1, N=4x^2\cos(y)+3x\cot(y)+5\sin(2y)$

$M_y=2x\cos(y), N_x=8x\cos(y)+3\cot(y)$

Check and find this is not exact

Then try to find integrating factor

$N_x-M_y=8x\cos(y)+3\cot(y)-2x\cos(y)=6x\cos(y)+3\cot(y)$

By observation, $\frac{N_x-M_y}{M}=3\cot(y)$

Therefore, the integrating factor is $\sin^{3}(y)$

$M'=2x\sin^4(y)+\sin^3(y)$

$\psi(x,y)=x^2 \sin^4(y)+x \sin^3(y)+h(y)$

$\psi_y=4x^2\sin^3(y)\cos(y) + x \sin^2(y)\cos(y)+h'(y)$

$h'(y)=10\sin^4(y) \cos(y)$

$h(y)=2\sin^5 (y)$

$\psi(x,y)=x^2 \sin^4(y)+x \sin^3(y)+2\sin^5 (y)$

**Jingyi Wang**:

Do not forget the function of Y. We also have to integrate N.

The final answer should be:

𝑥2sin4𝑦+𝑥sin3𝑦+2sin5y

**Jingze Wang**:

Sorry I have not finished my typed solution at that time, so you just saw part of my solution, but we get the same answer finally :)

**Jingyi Wang**:

Sorry about that. I did not see that part. Please ignore my post.

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