# Answer to Question #20825 in Linear Algebra for noor

Question #20825

solve the wave equation c^2 uxx = u tt for t ≥ 0 0 ≤ x ≤ π . Take c = 1 , take initial and boundary conditions

u(x , 0) = sin x and ut(x , 0) = 0

u( 0 , t) = u ( L , t ) =0 ?

u(x , 0) = sin x and ut(x , 0) = 0

u( 0 , t) = u ( L , t ) =0 ?

Expert's answer

solve the wave equation c^2 uxx = u tt for t ≥ 00 ≤ x ≤ π . Take c = 1 , take initial and boundary conditions

u(x , 0) = sin x and ut(x , 0) = 0

u( 0 , t) = u ( L , t ) =0 ?

U(x,t)=T(t)*X(x)

T''(t)/T(t)=X''(x)/X(x)=b

X''(x)+bX(x)=0

X(0)=0

X(L=π)=0

X(x)=Csin(nx) , b=1

T''(t)+T(t)=0

T(t)=Asin(kt)+Bcos(kt)

U(x,t)=Summa[n,k,{1, Infinity} (Aksin(kt)+Bkcos(kt))Cnsin(nx)]

U(x , 0) = sin x and Ut(x , 0) = 0

U(x , 0) = sin x = Summa[ (Aksin(0)+Bkcos(0))Cnsin(nx)]

BkCn=1 if n=k=1

Ut(x , 0) = 0= Summa[ (Akcos(0)-Bksin(0))Cnsin(nx)]

AkCn=0

Answer: U(x,t)=sin(x)+sin(t)

u(x , 0) = sin x and ut(x , 0) = 0

u( 0 , t) = u ( L , t ) =0 ?

U(x,t)=T(t)*X(x)

T''(t)/T(t)=X''(x)/X(x)=b

X''(x)+bX(x)=0

X(0)=0

X(L=π)=0

X(x)=Csin(nx) , b=1

T''(t)+T(t)=0

T(t)=Asin(kt)+Bcos(kt)

U(x,t)=Summa[n,k,{1, Infinity} (Aksin(kt)+Bkcos(kt))Cnsin(nx)]

U(x , 0) = sin x and Ut(x , 0) = 0

U(x , 0) = sin x = Summa[ (Aksin(0)+Bkcos(0))Cnsin(nx)]

BkCn=1 if n=k=1

Ut(x , 0) = 0= Summa[ (Akcos(0)-Bksin(0))Cnsin(nx)]

AkCn=0

Answer: U(x,t)=sin(x)+sin(t)

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