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 + x3.1pksri@iitp.ac.in