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9. Solve the Goursat problem: (U₁ - c²Uxx = 0 u|x-ct=0 = x² |u|x+c=0 = x². Hint: Use the formula for general solutions of wave equation on the real line. PS4 Q9: eplations First the geneal solution for wave we deduced in the lecture notes are Ucx,t) = Fcxt ct) + GCx-CE for any F, G Nert we me the boundary conditions to determine FG when x-ct=0, Narely t = £, we have x² = 4/x-44 = FCX+x)+ GCX-X) = F(x) FG(0) when x+ ct=0. Narely t=-*, we have 3 x4 = u\x+ of plugin x=0 in = Fcxx)+ GCx+x) = Frost Glax) , we get FCO)+ Gio)=0 Replace x by in @, we get FCX)=-G(o) replaces x by 1 in , we get So G(x) = - F (o) The solution is (CX)= Fcx+ot)+ GCX-cf) = (X+1+)2 - G(o) + (*c+)" -F(o) using ③. Frost G(0) = 0, 16 So (cx,t) = (x+ct)² + (ACE) 9. 4 +16 - (Frostt(0)) CX-CE)9 + 16 Scanned with CS CamScanner Scanned with CamScanner Solve the following Gourset problem Utt - Uxx = 0 U|x-t=0 = 4x² U|x+t=0 = 4x.

9. Solve the Goursat problem: (U₁ - c²Uxx = 0 u|x-ct=0 = x² |u|x+c=0 = x². Hint: Use the formula for general solutions of wave equation on the real line. PS4 Q9: eplations First the geneal solution for wave we deduced in the lecture notes are Ucx,t) = Fcxt ct) + GCx-CE for any F, G Nert we me the boundary conditions to determine FG when x-ct=0, Narely t = £, we have x² = 4/x-44 = FCX+x)+ GCX-X) = F(x) FG(0) when x+ ct=0. Narely t=-*, we have 3 x4 = u\x+ of plugin x=0 in = Fcxx)+ GCx+x) = Frost Glax) , we get FCO)+ Gio)=0 Replace x by in @, we get FCX)=-G(o) replaces x by 1 in , we get So G(x) = - F (o) The solution is (CX)= Fcx+ot)+ GCX-cf) = (X+1+)2 - G(o) + (*c+)" -F(o) using ③. Frost G(0) = 0, 16 So (cx,t) = (x+ct)² + (ACE) 9. 4 +16 - (Frostt(0)) CX-CE)9 + 16 Scanned with CS CamScanner Scanned with CamScanner Solve the following Gourset problem Utt - Uxx = 0 U|x-t=0 = 4x² U|x+t=0 = 4x.

College Algebra (MindTap Course List)
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter1: Equations And Inequalities
Section1.CT: Chapter Test
Problem 5CT: Solve for x: z=x.
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9. Solve the Goursat problem: (U₁ - c²Uxx = 0 u|x-ct=0 = x² |u|x+c=0 = x². Hint: Use the formula for general solutions of wave equation on the real line. PS4 Q9: eplations First the geneal solution for wave we deduced in the lecture notes are Ucx,t) = Fcxt ct) + GCx-CE for any F, G Nert we me the boundary conditions to determine FG when x-ct=0, Narely t = £, we have x² = 4/x-44 = FCX+x)+ GCX-X) = F(x) FG(0) when x+ ct=0. Narely t=-*, we have 3 x4 = u\x+ of plugin x=0 in = Fcxx)+ GCx+x) = Frost Glax) , we get FCO)+ Gio)=0 Replace x by in @, we get FCX)=-G(o) replaces x by 1 in , we get So G(x) = - F (o) The solution is (CX)= Fcx+ot)+ GCX-cf) = (X+1+)2 - G(o) + (*c+)" -F(o) using ③. Frost G(0) = 0, 16 So (cx,t) = (x+ct)² + (ACE) 9. 4 +16 - (Frostt(0)) CX-CE)9 + 16 Scanned with CS CamScanner Scanned with CamScanner
Transcribed Image Text:9. Solve the Goursat problem: (U₁ - c²Uxx = 0 u|x-ct=0 = x² |u|x+c=0 = x². Hint: Use the formula for general solutions of wave equation on the real line. PS4 Q9: eplations First the geneal solution for wave we deduced in the lecture notes are Ucx,t) = Fcxt ct) + GCx-CE for any F, G Nert we me the boundary conditions to determine FG when x-ct=0, Narely t = £, we have x² = 4/x-44 = FCX+x)+ GCX-X) = F(x) FG(0) when x+ ct=0. Narely t=-*, we have 3 x4 = u\x+ of plugin x=0 in = Fcxx)+ GCx+x) = Frost Glax) , we get FCO)+ Gio)=0 Replace x by in @, we get FCX)=-G(o) replaces x by 1 in , we get So G(x) = - F (o) The solution is (CX)= Fcx+ot)+ GCX-cf) = (X+1+)2 - G(o) + (*c+)" -F(o) using ③. Frost G(0) = 0, 16 So (cx,t) = (x+ct)² + (ACE) 9. 4 +16 - (Frostt(0)) CX-CE)9 + 16 Scanned with CS CamScanner Scanned with CamScanner
Solve the following Gourset problem Utt - Uxx = 0 U|x-t=0 = 4x² U|x+t=0 = 4x.
Transcribed Image Text:Solve the following Gourset problem Utt - Uxx = 0 U|x-t=0 = 4x² U|x+t=0 = 4x.
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