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Questions and Answers
Find the degree and order of $(\frac{d^2y}{dx^2})^{\frac{3}{2}} + 5 \frac{dy}{dx} + y = e^{x^2}$
Find the degree and order of $(\frac{d^2y}{dx^2})^{\frac{3}{2}} + 5 \frac{dy}{dx} + y = e^{x^2}$
$\frac{1}{D^2 + 2} cos 3x = ?$
$\frac{1}{D^2 + 2} cos 3x = ?$
If $x = u + v$ and $y = v$ then $\frac{\partial(x,y)}{\partial(u,v)}$ = ?
If $x = u + v$ and $y = v$ then $\frac{\partial(x,y)}{\partial(u,v)}$ = ?
$\int \int dxdy = ?$
$\int \int dxdy = ?$
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$PI = \frac{1}{D^2 - 2} sin 3x = ?$
$PI = \frac{1}{D^2 - 2} sin 3x = ?$
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$L[f(t) = e^{-at}]=?$
$L[f(t) = e^{-at}]=?$
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$L[t(t-1)] = ?$
$L[t(t-1)] = ?$
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$L = ?$
$L = ?$
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Study Notes
Mathematics II - END SEMESTER EXAMINATION
Instructions to the Candidate
- The exam duration is 2 hours
- Figures to the right indicate full marks
- Draw neat sketches and diagrams whenever necessary
Part - A
Multiple Choice Questions
- Answer any 10 questions (1 mark each)
Differential Equations
- The degree and order of (d2ydx2)32+5dydx+y=ex2(\frac{d^2y}{dx^2})^{\frac{3}{2}} + 5 \frac{dy}{dx} + y = e^{x^2}(dx2d2y)23+5dxdy+y=ex2 can be determined
- The options are 2,2; 2,3; 1,2; and 3,2
Inverse Laplace Transforms
- PI=1D2+2cos3xPI = \frac{1}{D^2 + 2} cos 3xPI=D2+21cos3x can be simplified to cos3x−7\frac{cos 3x}{-7}−7cos3x, sin3x−11\frac{sin 3x}{-11}−11sin3x, cos3x−9\frac{cos 3x}{-9}−9cos3x, or None of these
- PI=1D2−2sin3xPI = \frac{1}{D^2 - 2} sin 3xPI=D2−21sin3x can be simplified to sin3x10\frac{sin 3x}{10}10sin3x, sin3x−11\frac{sin 3x}{-11}−11sin3x, sin3x−7\frac{sin 3x}{-7}−7sin3x, or None of these
Partial Derivatives
- If x=u+vx = u + vx=u+v and y=vy = vy=v, then ∂(x,y)∂(u,v)\frac{\partial(x,y)}{\partial(u,v)}∂(u,v)∂(x,y) can be evaluated
- The options are 0, 1, 2, and None of these
Double Integrals
- The value of ∫∫dxdy\int \int dxdy∫∫dxdy is 1, 2, 3, or 4
- The value of ∫∫∫dxdydz\int \int \int dxdydz∫∫∫dxdydz is 1, 2, 3, or 4
Differential Equations - Integrating Factor
- If 1M(x,y)∂M(x,y)∂y=1N(x,y)∂N(x,y)∂x\frac{1}{M(x,y)} \frac{\partial M(x,y)}{\partial y} = \frac{1}{N(x,y)} \frac{\partial N(x,y)}{\partial x}M(x,y)1∂y∂M(x,y)=N(x,y)1∂x∂N(x,y), then e∫(∂M(x,y)∂y−∂N(x,y)∂x)dxe^{\int (\frac{\partial M(x,y)}{\partial y} - \frac{\partial N(x,y)}{\partial x}) dx}e∫(∂y∂M(x,y)−∂x∂N(x,y))dx is an integrating factor of the differential equation M(x,y)dx + N(x,y)dy = 0
- The statement is either True or False
Laplace Transforms
- L[f(t)=e−at]L[f(t) = e^{-at}]L[f(t)=e−at] can be evaluated
- The options are 1s\frac{1}{s}s1, 1s+a\frac{1}{s+a}s+a1, e−ats\frac{e^{-at}}{s}se−at, and 1s−a\frac{1}{s-a}s−a1
- L[t(t−1)]L[t(t-1)]L[t(t−1)] can be evaluated
- The options are 2s3\frac{2}{s^3}s32, ess2\frac{e^s}{s^2}s2es, 1s3\frac{1}{s^3}s31, and None of these
- LLL can be evaluated
- The options are 1s\frac{1}{s}s1, 1s2\frac{1}{s^2}s21, 1s3\frac{1}{s^3}s31, and None of these
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Description
This exam assesses the student's understanding of differential equations, covering topics such as degree and order of equations. It includes multiple-choice questions and requires neat sketches and diagrams.
