Tutorial ODE (EEM1026) trims 2 1718

1. State the order, linearity, homogeneous or non- homogeneous of following differential

equations.
4

(a) d 3 y + 5  dy   4y = ex (b) y''  5xy' + y = 3 sinx

 
dx 3  
dx

2. Solve the following differential equations.

x 1
(a) y’ = x2 - x (b) y'  x  1 (c) y” = 24x
Ans : (a) x3/3 – x2/2 +c, (b) x + 2lnx-1 + c or x -1 + 2lnx-1 + c (by using u=x-1),
(c) 4x3 + cx + d

3. Solve the following initial value problems.

(c) y '  x 1  y ; y(0) = 1
2
(a) y’ = y 2 + y – 6 ; y(5) = 10 (b) y '  y ln y , y(1) = e
x y
5 ( x 5 )
26  24e 4
Ans:(a) 13  8e 5( x 5) ; (b) y = ex, (c) y  1  x
4

4. By using the method of separation of variables, find the general solution of the
following differential equations:
(a) y’ = xy
2

(b) 1  x y'  2 y  (c) y 2 y’ = (y 3 +1)
Ans: (a) y  ce
x 2
/2
, (b) y  A
1 x 
, (c) y  Ae  1
3x

1/ 3

1 x

5. Show that the equation is exact, and obtain its general solution. Find the particular
solution corresponding to the given initial condition if given.
(a) [(x + 1)ex –ey] dx –xey dy = 0 (b) 3x2(y-1)2 dx + [2x3(y -1)+y] dy = 0; y(1) = 1/2
dy 1  e 2 y
3 3 2
(c) 2(x + y ) dx + 6xy dy = 0 (d) dx  2 xe 2 y
x  2y
(e) (x + tan-1y)dx + dy  0 (f) 2y2+ yexy + (4xy+ xexy+2y)y’ = 0
1 y 2

(g) x cos(2y – x) – sin(2y –x) –2x cos(2y – x)y’ = 0; y(/12) = /8

Ans:(a) x(ex –ey) = c, (b) x3(y –1)2 +(y2/2)= 3/8 (c) 2xy3+x4/2 = c ;
2 y
(d) x  xe  C ; (e) x tan-1y + ln1+y2 + x2/2 = C; (f) 2xy2+ exy + y2 = c;
(g)  x sin(2 y  x)   / 24

6. Newton's law of cooling states that dT  k (T - TS ) . Given that a hot bottle with initial
dt
temperature of 80 C is allowed to cool in air which remains at temperature 250C. It is
0

observed that in 20 minutes, the bottle has cooled down to 600C. What is the temperature
of the bottle after another half hour? [T(t) = 42.770C]

1
7. Prove that given function is an integrating factor. Not need to solve.
(a) 2ydx + xdy = 0, µ(x) = x. (b) y2 dx + (1 + xy) dy = 0, µ(x, y) = exy.
8. By using the formula  ( x)  e 
R ( x ) dx 1  M N 
where R( x)    or
N  y x 

 ( y)  e 
R ( y ) dy
with R( y)  1  N  M  . Find a suitable integrating factor µ(x) or µ(y),
M  x y 
and use it to find the general solution of the differential equation.
(a) (y + 1) dx - (x + 1) dy = 0 (b) (xy + y +1)dx + x dy = 0
(c) (x + y) dx + xlnx dy = 0 (d) 2  ln x  2 x 2 y dx + x3 dy = 0
(e) xn(x2 + y2 + x)dx + yxn+1dy = 0 (f) (xy3 + y2)dx + (y – xy + y3 cos y)dy = 0
y 1
, general solution is x  1  c ; (b) ex, general solution is xyex +ex = c ;
1
Ans: (a)
x  1 2

1 2 x 2 2 4 3
(c) ; y ln x + x = c; (d) ; x 2 y  2 ln x  (ln x)  C ; (e) n , x y  x  x  C .
1
x x 2 x 2 4 3
2
1 x 1 x
(f) 3 ;    sin y  c
y 2 y y

9. Find the general solutions of the following linear differential equations. If initial
condition is given, find the corresponding particular solution.
(a) 2xy’ + y = 10 x (b) cos x dy  y sin x  cos 2 x
dx
(c) y’ + y = ex, y(0) = 1 (d) (1+x)y’ + y = cos x, y(0) = 1, x > -1.
dy 1 dy
(e)  y  x, y(1)  5 [exam Feb 2015] (f)  y  2 sin x
dx x dx

, (b) y =xcosx + ccosx, (c) y  2 e  e  , (d) y  1  x sin x  1 ,

c 1 1
Ans: (a) y  5 x 
x x

x
2
(e) y  x  14 , (f) y  sin x  cos x  ce  x
3 3x

10. Solve the following Bernoulli equations:

1 y
(a) y' y  y , y(0)  2 (c) xy' y  x y (d) y' x  x y
2 2 2 2
(b) x2y’ + 2xy = 5y3
1/ 2
 x  2
, (b) y   2  cx5 
2 x
Ans: (a) y  1  3e , (c) y  1 , (d) y
   x 2  cx  x  cx
3

11. Find a general solution of the following homogeneous linear differential equations
with constant coefficients:
(a) y” + 7y = 0 (b) y”- 6y + 25y = 0 (c) y(iv) -29y” + 100y = 0

2
(d) Solve the following initial value problems:
d2y 
 2 y  0, y(0)  5, y' ( )  4,   0 [exam Feb 2015]
dx 2

Ans: (a) y = A cos 7 x +B sin 7 x, (b) y = e3x(Acos 4x+B sin 4x),
4
(c) y = Ae-2x+Be2x+Ce-5x+De5x, (d) y =  sin x + 5 cos x


12. Solve the following initial value problems:

(a) y” + 4y’ + 5y = 0, y(0) = -3, y’(0) = 0
(b) y”’-3y” + 3y’ - y = 0, y(0) = 2, y’(0) = 2, y”(0) = 10
(c) y(iv) + 3y”- 4y = 0, y(0) = 0, y’(0) = -10, y”(0) = 0, y”’(0) = 40
Ans: (a) y = -3e-2x(cos x + 2 sin x), (b) y = (2 + 4x2)ex, (c) y = -5 sin 2x

13. By using the method of undetermined coefficients, find the particular solution yp of
the following inhomogeneous differential equations:
(a) y” + 2y’ + 3y = 1 + xex (b) y” + y’ + y = sin2 x (c) y” – y’ - 6y = 2 sin 3x
d2y
(d)  9 y  12 x cos 3x (e) y” + y = cos2 x
dx 2
1 1
Ans: (a) 3   6 x  9 e , (b) 2  26 cos 2 x  13 sin 2 x , (c) 39 cos 3x  5 sin 3x  ,
1 x 1 3 1 1
 
x
y p  x 2 sin 3x  cos 3x, (e) yp = -(1/6)cos 2x + 1/2
(d) 3

14. Solve the following initial value problems:

(a) y” + 4y = 2x, y(0) = 1, y’(0) = 2 (b) y” + 9y = sin 2x, y(0) = 1, y’(0) = 0
(c) y” - 2y’ + 2y = x + 1, y(0) = 3, y’(0) = 0
3 1 2 1
Ans: (a) cos 2 x  4 sin 2 x  2 x , (b) cos 3x  15 sin 3x  5 sin 2 x ,
 5  1
(c) e  2 cos x  2 sin x   2 x  1
x

 

15. Solve the following problem:

2 x
(a) y”5y’ + 6y = -3sin2x [exam Feb 2016] (b) y"5 y'9 y  e 2 x

Ans: (a) y = c1 e2x+ c2 e3x + (-3/52)sin2x + (-15/52)cos 2x,

5 61 5 61
x x 1 2 x 1 13
(d) y  c1e 2
 c2 e 2
 e  x
15 9 81

3
 dy
16. Consider the solution of  (6 x  2) y  0 in the form of power series in x about
dx

x0 = 0, i.e., y   cn x n . Find the first five nonzero terms of this series solution.
n 0

[exam Feb 2015]

22 3 67
Ans: y  c0  2c0 x  5c0 x  c0 x  c0 x 4  
2

3 6


17. Consider a power series solution y   cn x of the differential equation

n

n 0
2
(1 + x )y” + 2xy’ - 2y = 0
 k 1 
(a) Show that c2 = c0, c3 = 0 and ck  2   k  1 ck , k  2
 
(b) Find the first five nonzero terms of the series solution.
1 1
Ans: y  co (1  x  3 x  5 x )  c1 x
2 4 6

18. Suppose that the linear differential equation

y” + x2y’ - 3xy = 0


has a power series solution y   c n x .

n

n 0

k 4
(a) Show that c2 = 0, c3 = ½ c0 and ck 2   (k  2)(k  1) ck 1 , k  2

1 3 1 4 1 7 1 10
(b) Show that y  co (1  x )  c1 ( x  x  x  x  )
2 6 252 5670

19. Consider the solution of y"5xy '  4 y  0 in the form of power series in x about

x0 = 0, i.e., y   cn x n . Find the first six coefficient terms of this series solution.
n 0

1 11
[exam Feb 2016] Ans: y  c0 c1x  2c0 x 2  c1 x3  c0 x 4  c1 x5  
6 120

20. Find the first six coefficient terms of the power series solution for the following
differential equation about x0 = 1.
y”+ xy = 0

Ans: y  c0 1  1 ( x  1) 2  1 x  13  1 x  14  1 x  15  

 2 6 24 30 

 c1 ( x  1)  x  13  x  14 
1 1 1
x  15  
 6 12 120 

4
 d 2 y dy
21. Consider the solution of   xy  0 in the form of power series in x about
dx 2 dx

x0 = 1, i.e., y  c
n 0
n ( x  1) n . Find the first six coefficient terms of this series solution.
Ans:

y   cn ( x  1) n  c0  c1 ( x  1)  c2 ( x  1) 2  c3 ( x  1)3  
n 0

c0 c c c c
 c0  c1 ( x  1)  ( x  1) 2  1 ( x  1) 2  0 ( x  1)3  0 ( x  1) 4  1 ( x  1) 4
2 2 3 24 8
c0 c
 ( x  1)5  1 ( x  1)5 
30 20
1 1 1 1
 c0 [1  ( x  1) 2  ( x  1)3  ( x  1) 4  ( x  1)5  ]
2 3 24 30
1 1 1
 c1[( x  1)  ( x  1) 2  ( x  1) 4  ( x  1)5 ]
2 8 20
end