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First Order Linear Differential Equations

Total questions: 10

Worksheet time: 12mins

Name
Class
Date
1.

Which of the following differential equations is not linear?

a)

 dydx+xy2=ex\frac{\text{d}y}{\text{d}x}+xy^2=e^x  

b)

 1xdydx−y=x+1\frac{1}{x}\frac{\text{d}y}{\text{d}x}-y=x+1  

c)

 xdydx+y=x−x2yx\frac{\text{d}y}{\text{d}x}+y=x-x^2y  

d)

 dydx=1−xy\frac{\text{d}y}{\text{d}x}=1-xy  

2.

The general solution of the differential equation


 dydx=2xy\frac{\text{d}y}{\text{d}x}=2xy 

 is

a)

 y=ex2+Ay=e^{x^2}+A  

b)

 y=Aex2y=Ae^{x^2}  

c)

 y=ex2y=e^{x^2}  

d)

 y=ln⁡(x2+A)y=\ln\left(x^2+A\right)  

3.

The particular solution of the differential equation


 dydx=y2e2x\frac{\text{d}y}{\text{d}x}=y^2e^{2x} 

for which y=1 when x=0 is given by

 

a)

 y=23−e2xy=\frac{2}{3-e^{2x}}  

b)

 y=21+e2xy=\frac{2}{1+e^{2x}}  

c)

 y=12−e2xy=\frac{1}{2-e^{2x}}  

d)

 y=e−2xy=e^{-2x}  

4.

An integrating factor of the differential equation


 dydx−2yx=3x2\frac{\text{d}y}{\text{d}x}-\frac{2y}{x}=3x^2 

is
 

a)

 e−2xe^{-2x}  

b)

 −ln⁡(x2)-\ln\left(x^2\right)  

c)

 x2x^2  

d)

 1x2\frac{1}{x^2}  

5.

An integrating factor of the differential equation


 (e2x+1)dydx+4e2xy=x\left(e^{2x}+1\right)\frac{\text{d}y}{\text{d}x}+4e^{2x}y=x 

is
 

a)

 e2x+1e^{2x}+1  

b)

 e2e2xe^{2e^{2x}}  

c)

 (e2x+1)2\left(e^{2x}+1\right)^2  

d)

 (e2x+1)4\left(e^{2x}+1\right)^4  

6.

The general solution of the differential equation


 dydx−(cot⁡x)y=sin⁡(2x)\frac{\text{d}y}{\text{d}x}-\left(\cot x\right)y=\sin\left(2x\right) 

 is

a)

 y=12sin⁡(2x)+Asin⁡xy=\frac{1}{2}\sin\left(2x\right)+A\sin x  

b)

 y=2sin⁡2(x)+sin⁡x+Ay=2\sin^2\left(x\right)+\sin x+A  

c)

 y=23sin⁡2(x)+Acosec⁡(x)y=\frac{2}{3}\sin^2\left(x\right)+A\operatorname{cosec}\left(x\right)  

d)

 y=2sin⁡2(x)+Asin⁡xy=2\sin^2\left(x\right)+A\sin x  

7.

The general solution of the differential equation


 xdydx+3y=x2x\frac{\text{d}y}{\text{d}x}+3y=x^2 

 is

a)

 y=x25+cx3y=\frac{x^2}{5}+\frac{c}{x^3}  

b)

 y=x2+Cxy=x^2+Cx  

c)

 y=x5+Cy=\frac{x}{5}+C  

d)

 y=x22+cxy=\frac{x^2}{2}+\frac{c}{x}  

8.

The particular solution of the differential equation


 dydx+2y=x\frac{\text{d}y}{\text{d}x}+2y=x 

for which y=0 when x=0 is given by

 

a)

 y=12x2e−2xy=\frac{1}{2}x^2e^{-2x}  

b)

 y=e−2x+2x−14y=\frac{e^{-2x}+2x-1}{4}  

c)

 y=1−e−2xy=1-e^{-2x}  

d)

 y=1−e−2x2y=\frac{1-e^{-2x}}{2}  

9.

The particular solution of the differential equation


 (x2+1)dydx+xy=x(x2+1)\left(x^2+1\right)\frac{\text{d}y}{\text{d}x}+xy=x\left(x^2+1\right) 

for which y=0 when x=0 is given by

 

a)

 y=13(x2+1)32−13y=\frac{1}{3}\left(x^2+1\right)^{\frac{3}{2}}-\frac{1}{3}  

b)

 y=x2(x2+2)4(x2+1)y=\frac{x^2\left(x^2+2\right)}{4\left(x^2+1\right)}  

c)

 y=13(x2+1)−13(x2+1)−12y=\frac{1}{3}\left(x^2+1\right)-\frac{1}{3}\left(x^2+1\right)^{-\frac{1}{2}}  

d)

 y=13(x2+1)+13(x2+1)12y=\frac{1}{3}\left(x^2+1\right)+\frac{1}{3}\left(x^2+1\right)^{\frac{1}{2}}  

10.

The general solution of the differential equation


 dydx+y=2e−x\frac{\text{d}y}{\text{d}x}+y=2e^{-x} 

 is

a)

 y=e−x(A+2x)y=e^{-x}\left(A+2x\right)  

b)

 y=Ae−x+2e−xy=Ae^{-x}+2e^{-x}  

c)

 y=Aex+2xe−xy=Ae^x+2xe^{-x}  

d)

 y=Ae−x+Bxe−xy=Ae^{-x}+Bxe^{-x}