1550078481-Ordinary_Differential_Equations__Roberts_

186 Ordinary Differential Equations

EXERCISES 4.1

In exercises 1-6 determine the largest interval on which the exis-

tence and uniqueness theorem guarantees the existence of a unique

solution for the given initial value problem.

l. 3y"-2y'+4y=x, y(-1)=2, y'(-1)=3

2. xy"'+xy'=4; y(l)=O, y'(l)=l, y"(l)=-1


3. x(x - 3)y" + 3y' = x^2 ; y(l) = 0, y'(l) = 1


4. x(x - 3)y" + 3y' = x^2 ; y(5) = 0, y'(5) = 1


5. v'f=Xy"-4y=sinx; y(-2)=3, y'(-2)=-l


6. (x^2 - 4)y" + (lnx)y = xex; y(l) = 1, y'(l) = 2


7. Verify that ex and e-x are both solutions of the differential equation
   y" - y = 0. Why are sinhx = (ex - e-x)/2 and coshx = (ex + e-x)/2
   also solutions?


8. Verify that the complex valued functions eix and e-ix are solutions of
   the differential equation y" + y = 0. Why are sin x = ( eix - e-ix) /2i and
   cosx = (eix + e-ix)/2 also solutions?


9. Verify that x, x-^2 , and c 1 x + c 2 x -^2 where c 1 and c2 are arbitrary
   constants are solutions of the differential equation x^2 y" + 2xy' - 2y = 0

for x > 0.

10. Verify that the functions 1 and x^2 are linearly independent on ( -oo, oo)

and are solutions of the differential equation 2yy" - (y')^2 = 0. Is the

li near combination y(x) = c 1 + c 2 x^2 a solution of this differential equa-

tion for arbitrary constants c 1 and c 2? Why does this not violate the

superposition theorem?

11. Show that the following sets of functions are linearly dependent on
    ( -oo, oo) by finding constants C1, c2, ... , Cn not all zero such that c 1 Y1 +
    C2Y2 + · · · + CnYn = 0.
    a. {2x,3x} b. {l,x,3x-4}

c. {l, sin^2 x, cos^2 x}

12. Use the Wronskian to show that the following sets of functions are

linearly independent on (-oo, oo ).

a. {sin x, cosx} b. {sin x, sin 2x}

c. {l, x, x^2 }