Question

The given two-parameter family is a solution of the indicated differential equation on the interval

(−[infinity],[infinity])(−[infinity],[infinity]).

y=c1​.ex. cosx+c2.​ex.sinx ; y′′−2y′+2y=0

Determine whether a member of the family can be found that satisfies the boundary conditions.

a. y(0)=1, y'(pi)=0
b. y(0)=1, y(pi)=-1
c. y(0)=1, y(pi/2)=1
d. y(0)=0, y(pi)=0

Answer 1

I do 1 option for you as an example, you need to check the leftover by yourself.

Explanation:

for d) y(0) = 0 and y'(pi) =0

`y(0) = C_1e^0cos(0)+ C_2 e^0 sin(0) = 0 \longrightarrow C_1 = 0`

`y(x) ' = C_1e^x cos(x) - C_1e^x sin (x) + C_2e^x sin(x) + C_2e^x cos(x)`

`y(\pi)'=C_1e^\pi cos(\pi)- C_1e^\pi sin(\pi)+ C_2e^\pi sin(\pi) + C_2e^\pi cos (\pi)`

Replace
`C_1 = 0` we have

`y'(\pi) = -C_2e^\pi = 0`

if and only if
`C_2 =0`

Hence the given solution does not work.

then, d is NOT the correct answer.