1550078481-Ordinary_Differential_Equations__Roberts_

28 Ordinary Differential Equations

6. Show that for all a :::; 0 and a ll b 2 0

l

(x - a)^3 , x < a :::; 0

Yab(x)= 0 , a:=;x:::;b

(x-b)^3 , o:::;b<x

is a solution of the initial value problem (20) y' = 3y^213 ; y(O) = 0 on

the interval (-00,00). This proves that the initia l value problem (20)

has an infinite number of solutions.

7. Verify that y = c 1 x lnx is a one-parameter family of solutions of the
   differential equation ( 21) ( x ln x )y' - ( 1 + ln x )y = 0 on the interval
   (0, oo). Solve, if you can, the three initial value problems consisting of
   the DE (21) and the following three sets of initial conditions

a. y(2) = 4 b. y(l)=O c. y(1)=2

8. Verify that y = c 1 e-x + c 2 e^2 x is a two-parameter family of solutions of

the differential equation (22) y" - y' - 2y = 0 on ( - oo, oo).

a. Solve the initial value problems consisting of the DE (22) and the

following two sets of initia l conditions.

(i) y(O) = 2, y'(O) = -5 (ii) y(l) = 3, y'(l) = -1

b. Solve the boundary value problems consisting of the DE (22) and

the following two sets of boundary conditions

(i) y(O) = 1, y(2) = 0 (ii) y(O) = 0, y' (2) = 1

9. Verify that y = c 1 x + c 2 x^2 + c 3 x^3 is a three-parameter family of solutions
   of the differential equation (23) x^3 y(^3 ) - 3x^2 y(^2 ) + 6xy(l) - 6y = O on
   (-oo, oo ). Find the solution of the initia l value problem consisting of
   the DE (23) and the initial conditions y(l) = 2, y' (1) = 3, y" (1) = 4.

10. Verify that y = c 1 x^2 + c 2 x^3 is a two-parameter family solutions of the

differential equation (24) x^2 y" - 4xy' + 6y = 0 on ( -oo, oo). Solve, if

you can, the boundary value problems consisting of the DE (24) and the

following sets of boundary conditions.

a. y(l) = 0, y(2) = -4

c. y(l) = 1, y'(2) = -12

e. y(O) = 0, y(2) = 4

b. y'(l) = 0, y(2) = 4

d. y'(1)=3, y'(2)=0

f. y(O) = 2, y'(2) = -1