Ans Sheet1

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NIIT University

MAT 101 Tutorial Sheet 1

Differential Equations

Answers

1. Determine whether 12 xy is a solution of .1)( 24 yy

Ans: LHS of the differential equation(DE) is positive and RHS is negative. No

function y(x) can satisfy this DE. The given DE has no solution.

2. Determine whetherxx xeexy 2)( is a solution of .02 yyy

Ans: It is a solution.

3. Is 1)( xy a solution of ?2 xyyy

Ans: It is not a solution.

4. Show that xy ln is a solution of ),0(0 onyyx but is not a solution on

.),(

Ans: xy ln is a solution on ),0( but it cannot be a solution on ),( , sincethe logarithm is undefined for negative numbers and zero.

5. Show that1

12

x

y is a solution of )1,1(02 2 onxyy but not on any

larger interval containing (-1, 1).

Ans: On (-1, 1)1

12

x

y and its derivative y are well defined functions. We

get1

12

x

y as a solution to the above DE on (-1, 1). However1

12 x

is not

defined at 1x and therefore could not be a solution on any interval containing

either of these two points.

6. Determine whether any of the functions (a) xy 2sin1 , (b) xy 2

or (c) xy 2sin2

13 is a solution to the initial value problem

.1)0(,0)0(;04 yyyy


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Ans: (a) xy 2sin1 is a solution to the DE and first condition, it does not satisfy

the second condition and so not a solution to the IVP.

(b) xy 2 satisfies both the initial conditions but not the DE. So it is not a

solution to the IVP.

(c) xy 2sin2

13 satisfies both the initial conditions and the DE. So it is a

solution to the IVP.

7. Find the solution to the to the initial value problem 2)3(;0 yyy

if the general solution to the differential equation is known to be xecxy 1)(

where c1 is an arbitrary constant.

Ans: .2)( 3 xeexy

8. Find a solution to the initial value problem

1)0(,0)0(;04 yyyy if the general solution to the differential

equation is known to be .2cos2sin)( 21 xcxcxy

Ans:xxy 2sin

2

1)(

is the solution to the IVP.

9. Find a solution to the boundary value problem

1)6

(,0)8

(;04

yyyy if the general solution to the differential

equation is known to be .2cos2sin)( 21 xcxcxy

Ans: ).2cos2(sin13

2)( xxxy

10. Determine c1 and c2 so that 12cos2sin)( 21 xcxcxy will satisfy the

conditions 0)8

(

y and .2)8

(

y

Ans: ).12(2

1)12(

2

121

candc


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11. Eliminate the arbitrary constants and obtain the differential equation

satisfied by it.

(a) 222 aqypx , where a is a fixed constant.

(b) xxx cebeaey 32 .

Ans: (a) .])(1[)( 3222 yya

(b) .06116 yyyy

12. Find all the values of m for whichy = emx

is a solution to the differential

equation .06116 yyyy

Ans: m = 1, 2, 3.

13. Reduce the following to first order equation and then solve ( set uy )

.0)( 3 yey y

Ans: .21 cycexy

14. Find the general solution of the following differential equations:

(a) .03612 yxyxx

(b) 21 yxeyy x .(c) dxxyxdyydx 33 .

Ans: (a) 32 )1(/ xxcy

(b) cexy x )1(1 2

(c) xxecy 33

15. Solve: (a) .1)2/(,sin2 2

rrddr

(b) .1)0(,11

x

dt

dxex

(c) .2)0(,1

yy

x

y

xy

Ans: (a)22

2sinln

r

(b) tx eee )1(1

(c) 28323232

xyy


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16. Solve using given substitutions:

(a) .,0cos)( tx

yx

x

yyyx

(b) .,2)(tan2222222

uyxxyyxyyx

Ans: (a) .lnsin cxx

y

(b) xceyx )sin( 22

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