Mathematics II - Differential Equations Exam

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+21​cos3x 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=D2−21​sin3x 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

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.