BIRLA INSTITUTE OF TECHNOLOGY AND SCIENCE, PILANI
MATHC 241: MATHEMATICS III
ASSIGNMENT-IV
17.09.2005
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1. Find the general solution in powers of x of ( x − 4) y"+3 xy '+ y = 0 .
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2. Find two linearly independent solutions of y"− xy '− x y = 0 .
3. For the following differential equation verify that the origin is a regular singular
point and find a Frobenius series solution corresponding to larger root of the
indicial equation.
4. 2 x 2 y"+3xy'−( x 2 + 1) y = 0 .
5. For the following differential equation verify that the origin is a regular singular
point and find a Frobenius series solution corresponding to larger root of the
indicial equation.
xy"+2 y '+ xy = 0
6. Use power series to solve the initial value problem
y′′ + xy′ − 2 y = 0, y (0) = 1, y′(0) = 0.
7. Determine the nature of the point x=0 for the differential equation
x 4 y′′ + ( x 2 sin x) y′ + (1 − cos x) y = 0
8. Find the exponents in the possible Frobenius series solutions of the equation near
x=0
2 x 2 (1 + x) y′′ + 3 x (1 + x)3 y′ − (1 − x 2 ) y = 0
9. Using the power series method, solve the differential equation
y′ − y = 0
10. Find the roots of the indicial equation of the differential equation
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x 2 y ′′ + x 2 + y = 0.
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11. Find the roots of the indicial equation of the differential equation
x 2 y ′′ + 3 xy ′ + y = 0.
12. For the differential equation
x 2 ( x 2 − 4) 2 y ′′ + x( x − 2) y ′ + y = 0 ,
locate and classify the singularities on the x – axis.
13. Solve the differential equation x 2 y ′ − y = −( x + 1) by power series method
14. ( )
Find the general solution in power of x of x 2 − 4 y ′′ + 3 xy ′ + y = 0
y ′′
15. Find the solution in power of x of +y=0
x
16. Solve 2 x 2 ( y ′′ + y ′) + xy ′ − y = 0
17. For the following equation, find a power series solution of the form ∑a n xn
y' = x 2 + y
18. For the following equation, find a power series solution of the form ∑a n xn
(1 − x ) y ' = x 2 − y
19. Find the general solution of
y = (1 + x 2 ) y ' '+ xy '
in terms of power series in x.
20. Give series solution to the equation
2 x 2 y ' '− xy '+ ( x 2 + 1) y = 0
21. Locate and classify the singular points of the differential equation
x 2 ( x 2 − 1) 2 y′′ + x ( x + 1) y′ − 2 y = 0
22. Determine the nature of the point x = 0 for the differential equation
x 2 y′′ + x3 y′ + (1 − cos x) y = 0
23. Find the indicial equation and its roots for the differential equation below
corresponding to the singular point x = 0.
( x + 1)
x y′′ + y′ − y = 0
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24. Find the first four nonzero terms in the power series expansion of the solution of
the initial value problem:
2 y′′ − y′ + ( x + 1) y = 0; y(0) = 0, y′(0) = 1
(Hint: The general term in the power series expansion is:
y ( n ) (0) n 1 1
x And y′′ = y′ − ( x + 1) y )
n! 2 2
25. Show that in any power series solution (in terms of powers of x) of the d.e.
i. y′ −3xy =0
there is no term containing x 3n −1 , n = 1, 2, …..
26. Determine the nature of the point x=0 for each of the following differential
equations:
xy’’+2xy’+3sinx y = 0
x3y’’- 3x2y’+sin(2x) y = 0
27. Locate and classify the singular points on the x-axis for
i. x2(4x2-1)2y’’+x(2x-1)y’+3y=0
28. Find a Frobenius series solution of the differential equation
i. 9x2y’’+3xy’+2(x-4)y=0
corresponding to larger root of its indicial equation. Write first non-zero four
terms of that solution.
29. Show that the differential equation
i. x2y’’+2x2y’+(x2+1/4 ) y =0
has only one Frobenius series solution. Find the general solution.