4.1PRELIMINARY THEORY-LINEAR EQUATIONS129ven in21. fi(1) = 1 + x (1) = x, fs(x) =xinitialiven in22. fi(1) = e", fa(x) = e, fax) = sinh xIn Problems 23-30 verify that the given functions form afundamental set of solutions of the differential equation onthe indicated interval. Form the general solution.= O for23. y"-y'- 12y = 0; e, e, (-0, )lution.24. y" – 4y = 0; cosh 2x, sinh 2x, (–x, x)25. y" – 2y' + 5y = 0; e cos 2x, e* sin 2x, (-, ∞)26. 4y" – 4y' + y = 0; ev2, xe², (-x, x)27. xy" – 6xy' + 12y = 0; x,x*. (0. x)ion oflitions28. xy" + xy' + y = 0; cos(ln x), sin(In x), (0, ×)solu-29. x'y" + 6x²y" + 4xy' – 4y = 0; x, x *,x² In x, (0, *)se thisndary30. y + y" = 0; 1,x, cos x, sin x. (-*, x)4.1.3 NONHOMOGENEOUS EQUATIONS) areon ofitionsIn Problems 31-34 verify that the given two-parameter fam-ily of functions is the general solution of the nonhomoge-neous differential equation on the indicated interval.31. y-7y' + 10y 24e:y= ce+ Ce + be". (-%, *)y is aterval32. y" +y = sec x:an beV=q Cos x+c sin x +x sin x + (cos x) In(cos x),

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Answered: 30. y+y" = 0; 1, x, cos x, sin x, (-x,…

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30. y+y" = 0; 1, x, cos x, sin x, (-x, x)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Question

4.1
PRELIMINARY THEORY-LINEAR EQUATIONS
129
ven in
21. fi(1) = 1 + x (1) = x, fs(x) =x
initial
iven in
22. fi(1) = e", fa(x) = e, fax) = sinh x
In Problems 23-30 verify that the given functions form a
fundamental set of solutions of the differential equation on
the indicated interval. Form the general solution.
= O for
23. y"-y'- 12y = 0; e, e, (-0, )
lution.
24. y" – 4y = 0; cosh 2x, sinh 2x, (–x, x)
25. y" – 2y' + 5y = 0; e cos 2x, e* sin 2x, (-, ∞)
26. 4y" – 4y' + y = 0; ev2, xe², (-x, x)
27. xy" – 6xy' + 12y = 0; x,x*. (0. x)
ion of
litions
28. xy" + xy' + y = 0; cos(ln x), sin(In x), (0, ×)
solu-
29. x'y" + 6x²y" + 4xy' – 4y = 0; x, x *,x² In x, (0, *)
se this
ndary
30. y + y" = 0; 1,x, cos x, sin x. (-*, x)
4.1.3 NONHOMOGENEOUS EQUATIONS
) are
on of
itions
In Problems 31-34 verify that the given two-parameter fam-
ily of functions is the general solution of the nonhomoge-
neous differential equation on the indicated interval.
31. y-7y' + 10y 24e:
y= ce+ Ce + be". (-%, *)
y is a
terval
32. y" +y = sec x:
an be
V=q Cos x+c sin x +x sin x + (cos x) In(cos x),

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