Use the family in Problem 1 to find a solution of y+y''=0 that satisfies the boundary conditions y(0)=0,y(1)=1.

Sinead Mcgee 2021-05-18 Answered

Use the family in Problem 1 to find a solution of y+y=0 that satisfies the boundary conditions y(0)=0,y(1)=1.

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Expert Answer

Ayesha Gomez
Answered 2021-05-19 Author has 104 answers

We have to find the solution of the given initial value problem, y+y=0, y(0)=0, y(1)=1
The differential equation can be written as, (D2+1)y=0.(1)
where D==ddx,D2==d2dx2
So the auxiliari equation of (1) is, D2+1=0D2=1D=+i
The required general solution is, y=c1cosx+c2sinx.(2)
Putting y=0 for x=0 in equation (2), we get, 0=c1cos0+c2sin0c1=0
And putting y=1 for x=1 in equation (2), we get, 1=c1cos1+c2sin1c2=1sin1[as c1=0]
Putting the values of c1 and c2 in equation (2), we get, y=(1sin1)sinx

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