Engineering Maths 2 - Jun 2015

First Year Engineering (P Cycle) (Semester 2)

TOTAL MARKS: 100
TOTAL TIME: 3 HOURS (1) Question 1 is compulsory.
(2) Attempt any four from the remaining questions.
(3) Assume data wherever required.
(4) Figures to the right indicate full marks. 1 (a) Solve $$4\dfrac{d^{4}y}{dx^{4}}-4\dfrac{d^{3}y}{dx^{3}}-23\dfrac{d^{2}y}{dx^{2}}+12\dfrac{dy}{dx}+36y=0$$(6 marks) 1 (a) Solve $$4\dfrac{d^{4}y}{dx^{4}}-4\dfrac{d^{3}y}{dx^{3}}-23\dfrac{d^{2}y}{dx^{2}}+12\dfrac{dy}{dx}+36y=0$$(6 marks) 1 (b) Solve $$\dfrac{d^{3}y}{dx^{3}}+6\dfrac{d^{2}y}{dx^{2}}+11\dfrac{dy}{dx}+6y=e^{x}+1$$ using inverse differential operator method(7 marks) 1 (b) Solve $$\dfrac{d^{3}y}{dx^{3}}+6\dfrac{d^{2}y}{dx^{2}}+11\dfrac{dy}{dx}+6y=e^{x}+1$$ using inverse differential operator method(7 marks) 1 (c) Solve (D²-2D)y=e^x sinx using method of undetermined coefficients(7 marks) 1 (c) Solve (D²-2D)y=e^x sinx using method of undetermined coefficients(7 marks) 10 (a) A periodic function f(t) with period 2 is defined by $$f(t)=\left\{\begin{matrix} t, &0\ltt\lt1 \\2-t,="" &1\ltt\lt2="" \end{matrix}\right.$$="" find="" l{f(t)}<="" a="">

</t<1>(6 marks) 10 (a) A periodic function f(t) with period 2 is defined by $$f(t)=\left\{\begin{matrix} t, &0\ltt\lt1 \\2-t,="" &1\ltt\lt2="" \end{matrix}\right.$$="" find="" l{f(t)}<="" a="">

</t<1>(6 marks) 10 (b) Find $$L^{-1}\left \{ \dfrac{5s-2}{3s^2+4s+8}+log\left ( \dfrac{1} {s^2}-1\right ) \right \}$$(7 marks) 10 (b) Find $$L^{-1}\left \{ \dfrac{5s-2}{3s^2+4s+8}+log\left ( \dfrac{1} {s^2}-1\right ) \right \}$$(7 marks) 10 (c) Solve using Laplace transform method $$\dfrac{d^2y}{dt^2}+2\dfrac{dy}{dt}+y=te^{-1}\ with \ y(0)=1,y^1(0)=2$$(7 marks) 10 (c) Solve using Laplace transform method $$\dfrac{d^2y}{dt^2}+2\dfrac{dy}{dt}+y=te^{-1}\ with \ y(0)=1,y^1(0)=2$$(7 marks) 2 (a) Solve (4D⁴-8D³-7D²+11D+6)y=0(6 marks) 2 (a) Solve (4D⁴-8D³-7D²+11D+6)y=0(6 marks) 2 (b) Solve (D²+4)y=x²+e^x using inverse differential operator method(7 marks) 2 (b) Solve (D²+4)y=x²+e^x using inverse differential operator method(7 marks) 2 (c) Solve (D²-2D+2)y=e^x tan x using method of variation of parameters(7 marks) 2 (c) Solve (D²-2D+2)y=e^x tan x using method of variation of parameters(7 marks) 3 (a) Solve $$\dfrac{dx}{dt}-7x+y=0,\dfrac{dy}{dt}-2x-5y=0$$(6 marks) 3 (a) Solve $$\dfrac{dx}{dt}-7x+y=0,\dfrac{dy}{dt}-2x-5y=0$$(6 marks) 3 (b) Solve $$x^{2}\dfrac{d^{2}y}{dx^{2}}+4x\dfrac{dy}{dx}+2y=e^{x}$$(7 marks) 3 (b) Solve $$x^{2}\dfrac{d^{2}y}{dx^{2}}+4x\dfrac{dy}{dx}+2y=e^{x}$$(7 marks) 3 (c) Solve y=2px+y²p³ by solving for x(7 marks) 3 (c) Solve y=2px+y²p³ by solving for x(7 marks) 4 (a) Solve $$(1+x)^{2}\dfrac{d^{2}y}{dx^{2}}+(1+x)\dfrac{dy}{dx}+y=2$$sin (log(1+x))$$\lt/span\gt\ltspan class='paper-ques-marks'\gt(6 marks)\lt/span\gt \lt/span\gt\ltspan class='paper-question'\gt\ltspan class='paper-ques-desc'\gt\ltb\gt4 (a)\lt/b\gt Solve $$(1+x)^{2}\dfrac{d^{2}y}{dx^{2}}+(1+x)\dfrac{dy}{dx}+y=2$$sin (log(1+x))$$(6 marks) 4 (b) Solve $$\dfrac{\mathrm dy}{\mathrm d x}-\dfrac{\mathrm dx}{\mathrm d y}=\dfrac{x}{y}-\dfrac{y}{x}$$ by solving for P.(6 marks) 4 (b) Solve $$\dfrac{\mathrm dy}{\mathrm d x}-\dfrac{\mathrm dx}{\mathrm d y}=\dfrac{x}{y}-\dfrac{y}{x}$$ by solving for P.(6 marks) 4 (c) Solve (px-y)(py+x)=a²p by reducing to Clairaut's form.(7 marks) 4 (c) Solve (px-y)(py+x)=a²p by reducing to Clairaut's form.(7 marks) 5 (a) From the function f(x²+y²,z-xy)=0 from the partial differential equation.(6 marks) 5 (a) From the function f(x²+y²,z-xy)=0 from the partial differential equation.(6 marks) 5 (b) Derive one dimensional wave equation as $$\frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}$$(7 marks) 5 (b) Derive one dimensional wave equation as $$\dfrac{\partial^2 u}{\partial t^2}=c^2\dfrac{\partial^2 u}{\partial x^2}$$(7 marks) 5 (c) Evaluate $$\int_{0}^{1}\limits\int_{x^2}^{2-x}\limits xy\ \mathrm d y \ \mathrm d x$$ by changing the order of integration(7 marks) 5 (c) Evaluate $$\int_{0}^{1}\limits\int_{x^2}^{2-x}\limits xy\ \mathrm d y \ \mathrm d x$$ by changing the order of integration(7 marks) 6 (a) Solve $$\frac{\partial^2u}{\partial x\partial y}=\sin x\ \sin y for \ which\ \frac{\partial u }{\partial y}=-2 \sin y$$when x=0 and u=0 when y is an odd multiple of $$\dfrac{\pi }{2}$$(6 marks) 6 (a) Solve $$\dfrac{\partial^2u}{\partial x\partial y}=\sin x\ \sin y for \ which\ \dfrac{\partial u }{\partial y}=-2 \sin y$$when x=0 and u=0 when y is an odd multiple of $$\dfrac{\pi }{2}$$(6 marks) 6 (b) Derive one dimensional heat equation as $$\dfrac{\partial u}{\partial t}=c^{2}\dfrac{\partial^2u }{\partial x^2}$$(7 marks) 6 (b) Derive one dimensional heat equation as $$\dfrac{\partial u}{\partial t}=c^{2}\dfrac{\partial^2u }{\partial x^2}$$(7 marks) 6 (c) Evaluate $$\int_{-1}^{1}\limits\int_{y}^{y}\limits\int_{x+y}^{x-y}\limits\ (x+y+z)\ dydxdz$$(7 marks) 6 (c) Evaluate $$\int_{-1}^{1}\limits\int_{y}^{y}\limits\int_{x+y}^{x-y}\limits\ (x+y+z)\ dydxdz$$(7 marks) 7 (a) Find the area between the parabolas y²=4ax and x²=4ay using double integral(6 marks) 7 (a) Find the area between the parabolas y²=4ax and x²=4ay using double integral(6 marks) 7 (b) Evaluate$$\int_{0}^{1} \limits\dfrac{dx}{\sqrt{1-x^{4}}}$$ using beta and gamma functions(7 marks) 7 (b) Evaluate$$\int_{0}^{1} \limits\dfrac{dx}{\sqrt{1-x^{4}}}$$ using beta and gamma functions(7 marks) 7 (c) Express the vector zi-2xj+yk in cylindrical coordinates(7 marks) 7 (c) Express the vector zi-2xj+yk in cylindrical coordinates(7 marks) 8 (a) Find the volume of the solid bounded by the planes x=0, y=0, x+y+z=1 and z=0 using triple integral(6 marks) 8 (a) Find the volume of the solid bounded by the planes x=0, y=0, x+y+z=1 and z=0 using triple integral(6 marks) 8 (b) Express $$\int_{0}^{\pi/2}\limits \sqrt{\sin \theta}\ d\theta \times \int_{0}^{\pi/2}\limits\dfrac{d\theta}{\sqrt{\sin \theta}}$$ using beta and gamma functions(7 marks) 8 (b) Express $$\int_{0}^{\pi/2}\limits \sqrt{\sin \theta}\ d\theta \times \int_{0}^{\pi/2}\limits\dfrac{d\theta}{\sqrt{\sin \theta}}$$ using beta and gamma functions(7 marks) 8 (c) Express the vector field 2yi-zj +3xk in spherical polar coordinate system(7 marks) 8 (c) Express the vector field 2yi-zj +3xk in spherical polar coordinate system(7 marks) 9 (a) Find Laplace transform of $$te^{-4t}\sin3t \ and \ \dfrac{e^u-e^{-u}}{t} $$(6 marks) 9 (a) Find Laplace transform of $$te^{-4t}\sin3t \ and \ \dfrac{e^u-e^{-u}}{t} $$(6 marks) 9 (b) Using f(t) in terms of unit step function and find its Laplace transform given that
$$\left\{\begin{matrix} t^2, &0\ltt \lt2\\="" 4t,="" &2\ltt="" \lt4="" \\8="" ,="" &t=""\gt4 \end{matrix}\right.$$</t>(7 marks) 9 (b) Using f(t) in terms of unit step function and find its Laplace transform given that
$$\left\{\begin{matrix} t^2, &0\ltt \lt2\\="" 4t,="" &2\ltt="" \lt4="" \\8="" ,="" &t=""\gt4 \end{matrix}\right.$$</t>(7 marks) 9 (c) Find $$L^{-1}\left \{ \dfrac{1}{(s+1)(s^2+9)} \right \}$$ using convolution theorem(7 marks) 9 (c) Find $$L^{-1}\left \{ \dfrac{1}{(s+1)(s^2+9)} \right \}$$ using convolution theorem(7 marks)