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Test:	DELA-Section 1.9-Exact equations
Number of Questions:	42
Time:

$\hspace{-6mm}\mbox{The differential equation}\\[2mm] M(x, y)dx+N(x, y)dy=0\\[2mm] \mbox{is said to be exact in a region} \ R \ \mbox{of the} \ xy\mbox{-plane if there exists a function} \ \phi(x, y) \ \mbox{such that}$

	$\dfrac{\partial\phi}{\partial x}=M, \ \dfrac{\partial\phi}{\partial y}=N \ \mbox{for all} \ (x, y) \ \mbox{in} \ \mathbb{R}.$
	$\dfrac{\partial\phi}{\partial y}=M, \ \dfrac{\partial\phi}{\partial x}=N \ \mbox{for all} \ (x, y) \ \mbox{in} \ \mathbb{R}.$
	$\mbox{None of the above}$
	$\dfrac{\partial M}{\partial x}=\phi, \ \dfrac{\partial N}{\partial y}=\phi \ \mbox{for all} \ (x, y) \ \mbox{in} \ \mathbb{R}.$
	$\dfrac{\partial M}{\partial y}=\phi, \ \dfrac{\partial N}{\partial x}=\phi \ \mbox{for all} \ (x, y) \ \mbox{in} \ \mathbb{R}.$

$\mbox{Solve} \ (2xy^2+3x^2)dx+(2x^2y+4y^3)dy=0.$

	$\mbox{None of the above}$
	$x^4+x^2y^2+y^3=c$
	$x^3+x^2y^2+y^4=c$
	$x^3+x^2y^2+y^3=c$
	$x^4 +x^2y^2+y^4=c$

$\mbox{Solve} \ ydx+(2xy-e^{-2y})dy=0, \ y>0.$

	$xe^{-2y}-\ln y=c$
	$xe^{-2y}+\ln y=c$
	$xe^{2y}+\ln y=c$
	$\mbox{None of the above}$
	$xe^{2y}-\ln y=c$

$\mbox{Solve} \ y^3dx+3xy^2dy=0.$

	$x/y^3=c$
	$x+y^3=c$
	$x-y^3=c$
	$\mbox{None of the above}$
	$xy^3=c$

$\mbox{Solve the initial-value problem} \ (3x^2 \ln x+x^2-y)dx-xdy=0, \ y(1)=5.$

	$y=\dfrac{x^3\ln x+5}{x^2}$
	$y=\dfrac{x^2\ln x+5}{x^2}$
	$y=\dfrac{x^2\ln x+5}{x}$
	$y=\dfrac{x^3\ln x+5}{x}$
	$\mbox{None of the above}$

$\mbox{Solve the initial-value problem} \ (4xy-3\sin x)dx+2x^2dy=0, \ y(2\pi)=0.$

	$y=\dfrac{3-3\cos x}{2x^2}$
	$y=\dfrac{2-2\cos x}{3x^2}$
	$y=\dfrac{3-3\cos x}{2x^3}$
	$\mbox{None of the above}$
	$y=\dfrac{2-2\cos x}{3x^3}$

$\mbox{Solve} \ (2xy+2x^3y-1)dx+(1+x^2)^2dy=0.$

	$y=\dfrac{\ln(1+x^2)}{1+x^2}+c$
	$y=\dfrac{\tan^{-1}x+c}{(1+x^2)^2}$
	$\mbox{None of the above}$
	$y=\dfrac{\ln(1+x^2)+c}{1+x^2}$
	$y=\dfrac{\tan^{-1}x+c}{1+x^2}$

$\mbox{Solve} \ (y^2-2x)dx+2xydy=0.$

	$x^2y^2-x^2=c$
	$xy^2-y^2=c$
	$xy^2-x^2=c$
	$x^2y-y^2=c$
	$\mbox{None of the above}$

$\mbox{Solve} \ (y-x^2)dx+2xdy=0, \ x>0.$

	$y=\dfrac{c}{\sqrt{x}}+\dfrac{x^2}{2}$
	$y=\dfrac{c}{\sqrt{x}}+\dfrac{x^2}{5}$
	$y=\dfrac{c}{\sqrt{x}}+\dfrac{x^2}{4}$
	$\mbox{None of the above}$
	$y=\dfrac{c}{\sqrt{x}}+\dfrac{x^2}{3}$

$\hspace{-6mm} \mbox{Solve} \ (2xy+\cos y)dx+(x^2-x\sin y-2y)dy=0.$

	$\mbox{None of the above}$
	$x^2y+x\cos y-y^2=c$
	$x^2y+x\cos y-y^3=c$
	$x^2y+xy\cos y-y^2=c$
	$x^3y+x\cos y-y^2=c$

$\hspace{-6mm} \mbox{Solve} \ (3xy+y^2)dx+(x^2+xy)dy=0.$

	$\dfrac{1}{3}x^3y+x^2y^2=c$
	$x^3y+\dfrac{1}{2}x^2y^2=c$
	$\dfrac{1}{3}x^3y+\dfrac{1}{2}x^2y^2=c$
	$x^3y+x^2y^2=c$
	$\mbox{None of the above}$

$\hspace{-16mm} \mbox{Solve} \ (6xy-y^3)dx+(4y+3x^2-3xy^2)dy=0.$

	$3x^3y-x^2y^2+2y^2=c$
	$\mbox{None of the above}$
	$3x^2y-xy^3+2y^2=c$
	$3x^2y-xy^2+2xy^3=c$
	$3x^3y-x^2y^2+2xy^3=c$

$\mbox{Solve} \ x^{-1}(xy-1)dx+y^{-1}(xy+1)dy=0.$

	$\mbox{None of the above}$
	$xy+\ln\|y/x\|=c$
	$y/x+\ln\|xy\|=c$
	$x/y+\ln\|xy\|=c$
	$xy+\ln\|x/y\|=c$

$\mbox{Solve} \ ydx-(2-x)dy=0, \ y\not = 0.$

	$y=\dfrac{2-x}{c}, \ c\not = 0$
	$y=c(2-x), \ c\not = 0$
	$y=\dfrac{c}{2-x}, \ c\not = 0$
	$\mbox{None of the above}$
	$y=\dfrac{1}{2-x}+c, \ c\not = 0$

$\mbox{Solve} \ 2xydx+(x^2+1)dy=0.$

	$(x^2+1)y=c$
	$\mbox{None of the above}$
	$(x^2+1)/y=c$
	$(x^2+1)y^2=c$
	$(x^2+1)xy=c$

$\mbox{Solve} \ (4e^{2x}+2xy-y^2)dx+(x-y)^2dy=0.$

	$6e^{2x}+3x^2y-3xy^2+y^3=c$
	$6e^{2x}+3x^2y+3xy^2+y^3=c$
	$6e^{2x}-3x^2y-3xy^2+y^3=c$
	$6e^{2x}-3x^2y-3xy^2-y^3=c$
	$6e^{2x}+3x^2y-3xy^2-y^3=c$
	$\mbox{None of the above}$

$\mbox{Solve} \ x^2ydx+y(x^3+e^{-3y}\sin y)dy=0.$

	$\dfrac{x^3e^{-3y}}{3}-\cos y=c$
	$\dfrac{x^3e^{3y}}{3}+\cos y=c$
	$\dfrac{x^3e^{-3y}}{3}+\cos y=c$
	$\mbox{None of the above}$
	$\dfrac{x^3e^{3y}}{3}-\cos y=c$

$\mbox{Solve} \ (2y^2+2y+4x^2)dx+(2xy+x)dy=0.$

	$x^2y^2+x^2y+x^4=c$
	$x^2y^2+x^2y+y^4=c$
	$\mbox{None of the above}$
	$xy^2+x^2y+y^4=c$
	$x^2y^2+xy^2+x^4=c$

$\mbox{Solve} \ (2xy-\sec^2x)dx+(x^2+2y)dy=0.$

	$\mbox{None of the above}$
	$x^2y-\cot x+y^2=c$
	$x^2y+\cot x+y^2=c$
	$x^2y+\tan x+y^2=c$
	$x^2y-\tan x+y^2=c$

$\mbox{Solve} \ dx+(x/y-\sin y)dy=0.$

	$xy+y\cos y-\sin y=c$
	$xy+x\cos y-\sin y=c$
	$\mbox{None of the above}$
	$xy+x\sin y-\cos y=c$
	$xy+y\sin y-\cos y=c$

$\mbox{Solve} \ (ye^{xy}+\cos x)dx+xe^{xy}dy=0, \ y(\pi/2)=0.$

	$\mbox{None of the above}$
	$y=\dfrac{\ln(1-\cos x)}{x^2}$
	$y=\dfrac{\ln(2-\sin x)}{x}$
	$y=\dfrac{\ln(1-\cos x)}{x}$
	$y=\dfrac{\ln(2-\sin x)}{x^2}$

$\mbox{Solve} \ (2x^2+y)dx+(x^2y-x)dy=0.$

	$x-yx^{-1}+\dfrac{y^2}{2}=c$
	$x-yx^{-2}+y^2=c$
	$2x-yx^{-2}+y^2=c$
	$2x-yx^{-1}+\dfrac{y^2}{2}=c$
	$\mbox{None of the above}$

$\mbox{Solve} \ (3x^2+y)dx+(x^2y-x)dy=0.$

	$x^2y^{-3}-y^{-2}=c$
	$x^2y^{-2}-y^{-1}=c$
	$xy^{-3}-y^{-1}=c$
	$x^2y^{-3}-y^{-1}=c$
	$\mbox{None of the above}$

$\mbox{Solve} \ \left (x^3+\dfrac{y}{x}\right )dx+(y^2+\ln x)dy=0.$

	$\dfrac{1}{4}x^4+\dfrac{1}{3}y^3-y\ln x=c$
	$\dfrac{1}{4}x^4+\dfrac{1}{3}y^3+y\ln x=c$
	$\dfrac{1}{4}x^4-\dfrac{1}{3}y^3-y\ln x=c$
	$\dfrac{1}{4}x^4-\dfrac{1}{3}y^3+y\ln x=c$
	$\mbox{None of the above}$

$\mbox{Solve} \ \left (\cos x+\ln y\right )dx+\left (\dfrac{x}{y}+e^y \right )dy=0.$

	$\mbox{None of the above}$
	$\sin x+x\ln y+e^y=c$
	$\sin x-x\ln y+e^y=c$
	$\sin x+x\ln y-e^y=c$
	$-\sin x+x\ln y+e^y=c$

$\mbox{Solve} \ \left (3xy+y^2\right )dx+\left (x^2+xy \right )dy=0.$

	$\mbox{None of the above}$
	$x^2y+\dfrac{1}{2}xy^2=c$
	$xy^2+\dfrac{1}{2}x^2y^2=c$
	$x^3y+\dfrac{1}{2}x^2y=c$
	$x^3y+\dfrac{1}{2}x^2y^2=c$

$\mbox{Solve} \ \left [\cos(xy)-xy\sin(xy)\right ]dx-x^2\sin(xy)dy=0.$

	$x\sin(xy)=c$
	$y\cos(xy)=c$
	$y\sin(xy)=c$
	$x\cos(xy)=c$
	$\mbox{None of the above}$

$\mbox{Solve} \ \left (3x^2y+2xy+y^3\right )dx+\left (x^2+y^2 \right )dy=0.$

	$e^{3x}y^2\left (\dfrac{1}{3}x^2+y^2\right )=c$
	$e^{3x}y^2\left (x^2+\dfrac{1}{3}y^2\right )=c$
	$e^{3x}y\left (x^2+\dfrac{1}{3}y^2\right )=c$
	$e^{3x}y\left (\dfrac{1}{3}x^2+y^2\right )=c$
	$\mbox{None of the above}$

$\mbox{Solve} \ (y^2+\cos x)dx+(2xy+\sin y)dy=0.$

	$x^2y+\sin x +\cos y=c$
	$x^2y+\sin x -\cos y=c$
	$xy^2+\sin x -\cos y=c$
	$xy^2+\sin x +\cos y=c$
	$x^2y-\sin x -\cos y=c$
	$xy^2-\sin x -\cos y=c$
	$\mbox{None of the above}$

$\mbox{Solve} \ xy[2\ln (xy)+1]dx+x^2dy=0, \ x,y>0.$

	$x^2\ln(xy)=c$
	$y^2\ln(xy)=c$
	$\mbox{None of the above}$
	$y\ln(xy)=c$
	$xy\ln(xy)=c$
	$x\ln(xy)=c$

$\mbox{Solve} \ [1+\ln(xy)]dx+xy^{-1}dy=0.$

	$\mbox{None of the above}$
	$xy\ln (xy)=c$
	$x\ln y=c$
	$xy\ln y=c$
	$x\ln (xy)=c$
	$y\ln (xy)=c$
	$xy\ln x=c$
	$y\ln x=c$

$\mbox{Solve} \ ydx-(2x+y^4)dy=0.$

	$x^{-2}y+y^2/2=c$
	$\mbox{None of the above}$
	$xy^{-2}+y^2/2=c$
	$xy^{-2}-y^2/2=c$
	$x^{-2}y-y^2/2=c$

$\mbox{Solve} \ [y\cos(xy)-\sin x]dx+x\cos(xy)dy=0.$

	$\mbox{None of the above}$
	$\sin(xy)+\sin x=c$
	$\sin(xy)+\cos x=c$
	$\cos(xy)+\sin x=c$
	$\cos(xy)+\cos x=c$

$\mbox{Solve} \ (xy-1)dx+x^2dy=0.$

	$xy+\ln\|x\|=c$
	$xy+\ln\|y\|=c$
	$xy-\ln\|x\|=c$
	$xy-\ln\|y\|=c$
	$\mbox{None of the above}$

$\mbox{Solve} \ \left (x^3+\dfrac{y}{x}\right )dx+\left (y^2+\ln x\right )dy=0.$

	$\dfrac{1}{4}x^4+\dfrac{1}{3}y^3+y\ln x=c$
	$3x^2+2y+\dfrac{\ln x}{y}=c$
	$\dfrac{1}{4}x^4+\dfrac{1}{3}y^3+x\ln x=c$
	$\mbox{None of the above}$
	$3x^2+2y+xy\ln x=c$

$\mbox{Solve} \ (x+2)\sin ydx+x\cos ydy=0.$

	$x^2+e^x+\sin y=c$
	$x^2e^x+\sin y=c$
	$x^2+e^x\sin y=c$
	$x^2e^x\sin y=c$
	$\mbox{None of the above}$

$\mbox{Solve} \ \left (1+ye^{xy}\right )dx+\left (2y+xe^{xy}\right )dy=0.$

	$x+xye^{xy}+y^2=c$
	$x+ye^{xy}+y^2=c$
	$\mbox{None of the above}$
	$x+e^{xy}+y^2=c$
	$x+xe^{xy}+y^2=c$

$\mbox{Solve} \ ydx+\left (2x-ye^y\right )dy=0.$

	$xy^2-2e^yy^2+2e^yy-e^y=c$
	$xy^2-2e^yy^2+e^yy-2e^y=c$
	$\mbox{None of the above}$
	$2xy^2-e^yy^2+2e^yy-e^y=c$
	$xy^2-e^yy^2+2e^yy-2e^y=c$

$\mbox{Solve} \ (y+3x^2)dx+xdy=0.$

	$xy+x^2=c$
	$x^3y+x=c$
	$\mbox{None of the above}$
	$xy+x^3=c$
	$x^2y+x^3=c$

$\mbox{Solve} \ (3xy-2y^{-1})dx+(x^2+xy^{-2})dy=0.$

	$x^2y-x^3y^{-1}=c$
	$\mbox{None of the above}$
	$x^3y-x^2y^{-1}=c$
	$x^3y-x^2y^{-2}=c$
	$x^2y-x^3y^{-2}=c$

$\mbox{Solve} \ e^xdx+(e^x\cot y+2y\csc y)dy=0.$

	$e^x\cos y+y^2=c$
	$\mbox{None of the above}$
	$e^x\sin y+xy^2=c$
	$e^x\sin y+y^2=c$
	$e^x\sin y+y=c$

$\mbox{Solve} \ (\sin y-2e^{-x}y\sin x)dx+(\cos y+2e^{-x}\cos x)dy=0.$

	$e^x\sin y+2y\cos x=c$
	$\mbox{None of the above}$
	$e^x\cos y+2y\sin x=c$
	$2e^x\cos y+y\sin x=c$
	$2e^x\sin y+y\cos x=c$

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