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MTH102: Assignment-3 1.D Reduce the following second order differential equation to first order differential equation and hence solve. (i) xy 00 + y 0 = y 02 (iii) yy 00 + y 02 +1=0 (iii) y 00 - 2y 0 coth x =0 2.T 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. 3.D Find the differential equation satisfied by each of the following two-parameter families of plane curves: (i) y = cos(ax + b) (ii) y = ax + b/x (iii) y = ae x + bxe x 4.D(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 5.T If p(x),q(x),r(x) are continuous functions on an interval I , then show that the set of solutions of the following linear homogeneous equation is a real vector space: y 00 + p(x)y 0 + q(x)y =0, x ∈I . (*) Also show that the set of solutions of the linear non-homogeneous equation y 00 +p(x)y 0 +q(x)y = r(x), x ∈I (#) is not a real vector space. Further, suppose y 1 (x),y 2 (x) are any two solutions of (#). Obtain conditions on the constants a and b so that ay 1 + by 2 is also its solution. 6.D Are the following functions linearly dependent on the given intervals? (i) sin 4x, cos 4x (-∞, ∞) (ii) ln x, ln x 3 (0, ∞) (iii) cos 2x, sin 2 x (0, ∞) (iv) x 3 ,x 2 |x| [-1, 1] 7.T(a) Show that a solution to (*) with x-axis as tangent at any point in I must be identically zero on I . (b) Let y 1 (x),y 2 (x) be two solutions of (*) with a common zero at any point in I . Show that y 1 ,y 2 are linearly dependent on I . (c) Show that y = x and y = sin x are not a pair solutions of equation (*), where p(x),q(x) are continuous functions on I =(-∞, ∞). 8.D(a) Let y 1 (x),y 2 (x) be two twice continuously differentiable functions on an interval I . Sup- pose that the Wronskian W (y 1 ,y 2 ) does not vanish anywhere in I . Show that there exists unique p(x),q(x) on I such that (*) has y 1 ,y 2 as fundamental solutions. (b) Construct equations of the form (*) from the following pairs of solutions: (i) e -x , xe -x (ii) e -x sin 2x, e -x cos 2x 9.T Let y 1 (x),y 2 (x) are two linearly independent solutions of (*). Show that (i) between consecutive zeros of y 1 , there exists a unique zero of y 2 ; (ii) φ(x)= αy 1 (x)+ βy 2 (x) and ψ(x)= γy 1 (x)+ δy 2 (x) are two linearly independent solutions of (*) iff αδ 6= βγ .

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MTH102: Assignment-3

1.D Reduce the following second order differential equation to first order differential equation

and hence solve.

(i) xy + y = y2 (iii) yy + y2 + 1 = 0 (iii) y 2y cothx = 0

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

tangent at the origin.

3.D Find the differential equation satisfied by each of the following two-parameter families

of plane curves:

(i) y = cos(ax+ b) (ii) y = ax+ b/x (iii) y = aex + bxex

4.D(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

5.T If p(x), q(x), r(x) are continuous functions on an interval I, then show that the set ofsolutions of the following linear homogeneous equation is a real vector space:

y+ p(x)y+ q(x)y = 0, x I. (*)Also show that the set of solutions of the linear non-homogeneous equation

y+p(x)y+q(x)y = r(x), x I (#)is not a real vector space. Further, suppose y1(x), y2(x) are any two solutions of (#).

Obtain conditions on the constants a and b so that ay1 + by2 is also its solution.

6.D Are the following functions linearly dependent on the given intervals?

(i) sin 4x, cos 4x (,) (ii) lnx, lnx3 (0,)(iii) cos 2x, sin2 x (0,) (iv) x3, x2|x| [1, 1]

7.T(a) Show that a solution to (*) with x-axis as tangent at any point in I must be identicallyzero on I.

(b) Let y1(x), y2(x) be two solutions of (*) with a common zero at any point in I. Showthat y1, y2 are linearly dependent on I.

(c) Show that y = x and y = sinx are not a pair solutions of equation (*), where p(x), q(x)

are continuous functions on I = (,).

8.D(a) Let y1(x), y2(x) be two twice continuously differentiable functions on an interval I. Sup-pose that the Wronskian W (y1, y2) does not vanish anywhere in I. Show that thereexists unique p(x), q(x) on I such that (*) has y1, y2 as fundamental solutions.

(b) Construct equations of the form (*) from the following pairs of solutions:

(i) ex, xex (ii) ex sin 2x, ex cos 2x

9.T Let y1(x), y2(x) are two linearly independent solutions of (*). Show that

(i) between consecutive zeros of y1, there exists a unique zero of y2;

(ii) (x) = y1(x) + y2(x) and (x) = y1(x) + y2(x) are two linearly independent

solutions of (*) iff 6= .