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Indian Institute of Technology Patna Practice Set— SE503: 2014-15 1 Q1. Classify the following Differential equation: Linear/ Non-linear/ Ordinary/ Partial etc. and specify the order. (i) y 00 +3y 0 + 20y = e x , (ii) p 1+ y 0 3 = x 2 , (iii) y sec +y 2 = cos x, (iv) y 0 + xy = cos y 0 , (v) (xy 0 ) 0 = xy, (vi) u x + u y = 0, (vii) u xx + u yy = u t . Q2. Verify that y = - 1 x+c is general solution of y 00 = y 2 . Find particular solutions such that (i) y(0) = 1, and (ii) y(0) = -1. In both cases, Find the largest interval I on which y is defined. Q3. Let V be a linear space of all twice differentiable functions with usual operations. Show that solutions of the differential equation y 00 + αy 0 + βy = 0 form a linear space. Q4. Solve the following linear first order differential equations using the method of variation of parameters: (i) xy 0 - 2y = x 4 (ii) y 0 + (cos x)y = sin x cos x Q5. Find the curve y = y(x) passing through origin for which y 00 = y 0 and the line y = x is tangent at the origin. Q6. (a) Find the values of m such that y = e mx is a solution of (i) y 00 +3y 0 +2y =0, (ii) y 00 - 4y 0 +4y =0, (iii) y 000 - 2y 00 - y 0 +2y =0. (b) Find the values of m such that y = x m (x> 0) is a solution of (i) x 2 y 00 - 4xy 0 +4y =0, (ii) x 2 y 00 - 3xy 0 - 5y =0. Q7. Find general solution of the following differential equations given a known solution y 1 : (i) x(1 - x)y 00 + 2(1 - 2x)y 0 - 2y =0, y 1 =1/x, (ii) (1 - x 2 )y 00 - 2xy 0 +2y =0, y 1 = x. Q8. Solve the following differential equations: (i) y 00 - 4y 0 +3y =0, (ii) y 00 +2y 0 +(ω 2 + 1)y =0, ω is real, (iii) 4y 00 - 12y 0 +9y =0. Q9. Solve the following Cauchy-Euler equations: (i) x 2 y 00 +2xy 0 - 12y =0, (ii) x 2 y 00 + xy 0 + y =0, (iii) x 2 y 00 - xy 0 + y =0. Q10. Solve the following differential equations: (i) y 000 - 8y =0, (ii) y (4) + y =0, (iii) y 000 - 3y 0 - 2y =0, (iv) y 000 - 6y 00 + 11y 0 - 6y =0. Q11. Find the particular solution for following equation using method of undetermined coefficients: (i) y 00 - y 0 - 6y = 20e -2x , (ii) y 00 - 2y 0 +5y = 25x 2 + 12, (iii) y 00 - 2y 0 + y =6e x , (iv) y 00 + y 0 = 10x 4 +2 (v) y 00 - 3y 0 +2y = 14 sin 2x - 18 cos 2x, Q 12. Solve the following equations using Method of variation of parameter: (i) y 00 +4y = tan 2x, (ii) y 00 +2y 0 +5y = e -x sec 2x, (iii) (x 2 - 1)y 00 - 2xy 0 +2y =(x 2 - 1) 2 (iv) y 00 + y = sec x csc x (v) x 2 y 00 - 2xy 0 +2y = xe -x , (vi) y 000 + y 00 + y 0 + y =1, (Hint for part (vi): Here homogeneous equation has solutions: cos x, sin x&e -x hence y p = v 1 (x) cos x + v 2 (x) sin x + v 3 (x)e -x . Now find v 1 v 2 &v 3 !) Q13. If y 1 (x) is solution of y 00 +P (x)y 0 +Q(x)y = R 1 (x) and y 2 (x) is solution of y 00 +P (x)y 0 +Q(x)y = R 2 (x), then show that y(x)= y 1 (x)+ y 2 (x) is a solution of y 00 + P (x)y 0 + Q(x)y = R 1 (x)+ R 2 (x). Using this result solve following differential equation: y 00 - 4y = e 2x + x 3 . 1 [email protected]

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Problem solving for Higher Engineering Mathematics

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Indian Institute of Technology PatnaPractice Set SE503: 2014-151

Q1. Classify the following Differential equation: Linear/ Non-linear/ Ordinary/ Partial etc. andspecify the order.

(i) y + 3y + 20y = ex, (ii)

1 + y3 = x2, (iii) y sec +y2 = cosx,(iv) y + xy = cos y, (v) (xy) = xy, (vi) ux + uy = 0, (vii) uxx + uyy = ut.

Q2. Verify that y = 1x+c is general solution of y = y2. Find particular solutions such that (i)y(0) = 1, and (ii) y(0) = 1. In both cases, Find the largest interval I on which y is defined.

Q3. Let V be a linear space of all twice differentiable functions with usual operations. Show thatsolutions of the differential equation y + y + y = 0 form a linear space.

Q4. Solve the following linear first order differential equations using the method of variation ofparameters:(i) xy 2y = x4 (ii) y + (cosx)y = sinx cosx

Q5. Find the curve y = y(x) passing through origin for which y = y and the line y = x is tangentat the origin.

Q6. (a) Find the values of m such that y = emx is a solution of(i) y + 3y + 2y = 0, (ii) y 4y + 4y = 0, (iii) y 2y y + 2y = 0.

(b) Find the values of m such that y = xm(x > 0) is a solution of(i) x2y 4xy + 4y = 0, (ii) x2y 3xy 5y = 0.

Q7. Find general solution of the following differential equations given a known solution y1:(i) x(1 x)y + 2(1 2x)y 2y = 0, y1 = 1/x, (ii) (1 x2)y 2xy + 2y = 0, y1 = x.

Q8. Solve the following differential equations:(i) y 4y + 3y = 0, (ii) y + 2y + (2 + 1)y = 0, is real, (iii) 4y 12y + 9y = 0.

Q9. Solve the following Cauchy-Euler equations:(i) x2y + 2xy 12y = 0, (ii) x2y + xy + y = 0, (iii) x2y xy + y = 0.

Q10. Solve the following differential equations:(i) y 8y = 0, (ii) y(4) + y = 0, (iii) y 3y 2y = 0, (iv) y 6y + 11y 6y = 0.

Q11. Find the particular solution for following equation using method of undetermined coefficients:(i) y y 6y = 20e2x, (ii) y 2y + 5y = 25x2 + 12, (iii) y 2y + y = 6ex,(iv) y + y = 10x4 + 2 (v) y 3y + 2y = 14 sin 2x 18 cos 2x,

Q 12. Solve the following equations using Method of variation of parameter:(i) y + 4y = tan 2x, (ii) y + 2y + 5y = ex sec 2x, (iii) (x2 1)y 2xy + 2y = (x2 1)2(iv) y + y = secx cscx (v) x2y 2xy + 2y = xex, (vi) y + y + y + y = 1,

(Hint for part (vi): Here homogeneous equation has solutions: cosx, sinx&ex hence yp =v1(x) cosx+ v2(x) sinx+ v3(x)e

x. Now find v1v2&v3!)

Q13. If y1(x) is solution of y+P (x)y+Q(x)y = R1(x) and y2(x) is solution of y+P (x)y+Q(x)y =

R2(x), then show that y(x) = y1(x) + y2(x) is a solution of y + P (x)y +Q(x)y = R1(x) +R2(x).

Using this result solve following differential equation: y 4y = e2x + [email protected]