Download GTU BE/B.Tech 2019 Summer 1st And 2nd Sem (New And SPFU) MTH002 Ordinary Differential Equation Question Paper

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GUJARAT TECHNOLOGICAL UNIVERSITY

BE - SEMESTER-I &II (SPFU) EXAMINATION — SUMMER-2019

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Subject Code: MTH002 Date: 01/06/2019

Subject Name: Ordinary Differential Equation

Time: 10:30 AM TO 01:00 PM Total Marks: 70

Instructions:

1. Attempt any five questions.

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2. Make suitable assumptions wherever necessary.
3. Figures to the right indicate full marks.

Q.1

(a) I Form the Differential equation from the solution y = c1 cosx + c2 sinx .

II. Verify that y = e^x(a cos x + b sinx) is a solution of y" — 2y’ + 2y = 0.

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(b) I Solve 2xy dx + x² dy = 0.

II. Solve the I.V.P.: xy' +y =0, y(2) = -2

Q.2

(a) Test the exactness and solve [(x+1)e^x —e^y]dx —xe^ydy=0; y(1) =0

(b) I Find the Differential equation of the orthogonal trajectory to y = cx².

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II. Find the orthogonal trajectories of the family of Cardioids r = a(1 + cos ?) , where a is parameter.

Q.3

(a) Solve the I.V.P.: dy/dx +y=x; y(0)=0

(b) Solve xy' =y²+y

Q.4

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(a) Verify that the functions x^-2 and x^-1 form a basis of solutions of 4x²y" — 3y = 0. Also write general solution of the given equation.

(b) I f(x)=e^x and g(x) = e ^-x are linearly independent or dependent?

II. If y = e^x(c1 cos x + c2 sin x) find the Wronskian W (y1, y2).

III. Find the General Solution of y"' +y'—2y =0.

Q.5

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(a) Solve the I.V.P.: y” +4y =8e^-2x +4x²+2; y(0) =2, y'(0) = 2.

(b) Using method of undetermined coefficients, solve y" + 4y = 8 x².

Q.6

(a) Solve y" —3y'+2y=e^x

(b) Find the roots of the indicial equation to x²y" + xy' — (2 —x)y =0 by Frobenius method.

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Q.7

(a) Find the series solution of (x² + 1)y” + xy' —xy = 0 near x = 0 .

(b) Find the series solution of y" = 2y' in power of x.

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