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Ashok Godase

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Mathematics

University

Differential Equations

Ashok Godase

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10 questions

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1. Multiple Choice
5 minutes
1 pt
The complementary function of the differential equation $\left(D^2-4\right)=0$ is
$ae^{2x}+be^{-2x}$
$ae^{4x}+be^{-4x}$
$\left(a+bx\right)e^{2x}$
$e^{-2x}\left(\cos2x+\sin2x\right)$
2. Multiple Choice
5 minutes
1 pt
For the differential equation $\left(D^3+3D^2+3D+1\right)y=e^{-x}$ the particular integral is given by
$-\frac{xe^{-x}}{3!}$
$\frac{xe^{-x}}{3!}$
$\frac{x^3e^{-x}}{3!}$
$\frac{e^{-x}}{3!}$
3. Multiple Choice
5 minutes
1 pt
A homogeneous linear differential equation $x^n\frac{d^ny}{dx^n}+p_1x^{n-1}\frac{d^{n-1}y}{dx^{n-1}}+..........+p_{n-1}x\frac{dy}{dx}+p_ny=X$ Where $p_1,p_2,........,p_n\$ are constants and X is a function of x can be transformed to a linear equation with constant coefficients by the substitution
$z=x$
$z=e^x$
$z=\log_ex$
$z=\frac{1}{x}$
4. Multiple Choice
5 minutes
1 pt
The particular integral of $\left(D^2-9\right)y=\cos3x$ is
$-\frac{\cos3x}{6}$
$-\frac{\cos3x}{18}$
$-\frac{\cos3x}{9}$
$-\frac{\cos3x}{3}$
5. Multiple Choice
5 minutes
1 pt
Which of the following statements are true?
a) $L\left(x^n\right)=\int_0^{\infty}e^{-sx}x^ndx$
b) $L\left(x^n\right)=\frac{\Gamma\left(n\right)}{s^{n+1}}$ , where n is a positive integer

c) $L\left(x^n\right)=\frac{n!}{s^{n+1}}$ , Where n is a positive integer

d) $L\left(e^{nx}\right)=\frac{1}{s-n}\$ if s-n<0
a, b and d are true
a and b are true
a and c are true
only b is true
6. Multiple Choice
5 minutes
1 pt
If $L\left(x\right)=\frac{1}{s^2}$ then $L^{-1}\left(\frac{1}{s^2}\right)$ is
1
$\frac{1}{x}$
$x^2$
$x$
7. Multiple Choice
30 seconds
1 pt
A partial differential equation which is linear in p and q which is also known as Langrange's equation is of the form
$\frac{P}{p}=\frac{Q}{q}=R$
$Pq+Qp=R$
$Pp+Qq=R$
$\frac{P}{q}+\frac{Q}{p}=R$
8. Multiple Choice
5 minutes
1 pt
The auxiliary equation of Lagrange's equation is
$Pdx+Qdy=R$
$\frac{dx}{P}=\frac{dy}{Q}=\frac{dz}{R}$
$Qdx=Pdy=Rdz$
$P=Q=R$
9. Multiple Choice
5 minutes
1 pt
The general solution of the partial differential equation 2p+3q=1 is
$\phi\left(3x-2y,y+3z\right)=0$
$\phi\left(3x+2y,y+3z\right)=0$
$\phi\left(3x-2y,y-3z\right)=0$
$\phi\left(3x+2y,y-3z\right)=0$
10. Multiple Choice
5 minutes
1 pt
The partial differential equation by eliminating the arbitrary constants a and b from $z=\left(x+a\right)\left(y+b\right)$ is
$z=p-q$
$z=\frac{p}{q}$
$z=pq$
$z=p+q$
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Differential Equations

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