Laplace Transforms and Integral Equations

1 Logo1 Transforms and New FormulasAn ExampleDouble CheckLaplace Transforms and IntegralEquationsBernd Schr oderBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace Transforms stays the Domain (t)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace Transforms stays the Domain (t)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace Transforms stays the Domain (t)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace Transforms stays the Domain (t)

2 OriginalDE & IVPB ernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace Transforms stays the Domain (t)OriginalDE & IVP-LBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace Transforms stays the Domain (t)OriginalDE & IVPA lgebraic equation forthe Laplace transform -LBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace Transforms stays the Domain (t) transform domain (s)OriginalDE & IVPA lgebraic equation forthe Laplace transform -LBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace Transforms stays the Domain (t)

3 transform domain (s)OriginalDE & IVPA lgebraic equation forthe Laplace transform -LAlgebraic solution,partial fractions?Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace Transforms stays the Domain (t) transform domain (s)OriginalDE & IVPA lgebraic equation forthe Laplace transformLaplace transformof the solution-LAlgebraic solution,partial fractions?Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace Transforms stays the Domain (t) transform domain (s)OriginalDE & IVPA lgebraic equation forthe Laplace transformLaplace transformof the solution- LL 1 Algebraic solution,partial fractions?

4 Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckEverything Remains As It WasNo matter what functions arise, the idea for solving differentialequations with Laplace Transforms stays the Domain (t) transform domain (s)OriginalDE & IVPA lgebraic equation forthe Laplace transformLaplace transformof the solutionSolution- LL 1 Algebraic solution,partial fractions?Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckThe Laplace transform of an integrals of the form t0f( )d arise in circuittheory: The charge of a capacitor is the Integral of thecurrent over time.

5 (We assume the capacitor is initially uncharged.) { t0f( )d }=F(s)sBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckThe Laplace transform of an integrals of the form t0f( )d arise in circuittheory: The charge of a capacitor is the Integral of thecurrent over time.(We assume the capacitor is initially uncharged.) { t0f( )d }=F(s)sBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckThe Laplace transform of an integrals of the form t0f( )d arise in circuittheory: The charge of a capacitor is the Integral of thecurrent over time.(We assume the capacitor is initially uncharged.)

6 { t0f( )d }=F(s)sBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckThe Laplace transform of an integrals of the form t0f( )d arise in circuittheory: The charge of a capacitor is the Integral of thecurrent over time.(We assume the capacitor is initially uncharged.) { t0f( )d }=F(s)sBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolve the Initial Value Problemf (t)+f(t) 2 t0f(z)dz=t,f(0)=0f (t)+f(t) 2 t0f(z)dz=t,f(0)=0sF+F 2Fs=1s2F(s+1 2s)=1s2Fs2+s 2s=1s2F=ss2(s 1)(s+2)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolve the Initial Value Problemf (t)+f(t) 2 t0f(z)dz=t,f(0)=0f (t)+f(t) 2 t0f(z)dz=t,f(0)=0sF+F 2Fs=1s2F(s+1 2s)=1s2Fs2+s 2s=1s2F=ss2(s 1)(s+2)

7 Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolve the Initial Value Problemf (t)+f(t) 2 t0f(z)dz=t,f(0)=0f (t)+f(t) 2 t0f(z)dz=t,f(0)=0sF+F 2Fs=1s2F(s+1 2s)=1s2Fs2+s 2s=1s2F=ss2(s 1)(s+2)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolve the Initial Value Problemf (t)+f(t) 2 t0f(z)dz=t,f(0)=0f (t)+f(t) 2 t0f(z)dz=t,f(0)=0sF+F 2Fs=1s2F(s+1 2s)=1s2Fs2+s 2s=1s2F=ss2(s 1)(s+2)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolve the Initial Value Problemf (t)+f(t) 2 t0f(z)dz=t,f(0)=0f (t)+f(t) 2 t0f(z)dz=t,f(0)=0sF+F 2Fs=1s2F(s+1 2s)=1s2Fs2+s 2s=1s2F=ss2(s 1)(s+2)Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolve the Initial Value Problemf (t)+f(t) 2 t0f(z)dz=t,f(0)=0f (t)+f(t) 2 t0f(z)dz=t,f(0)=0sF+F 2Fs=1s2F(s+1 2s)=1s2Fs2+s 2s=1s2F=ss2(s 1)(s+2)

8 Bernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolvef (t)+f(t) 2 t0f(z)dz=t,f(0)=0F=ss2(s 1)(s+2)=As+Bs2+Cs 1+Ds+2s=As(s 1)(s+2)+B(s 1)(s+2)+Cs2(s+2)+Ds2(s 1)s=0 :0=B ( 1) 2,B=0s=1 :1=C 12 3,C=13s= 2 : 2=D ( 2)2 ( 3),D=16s= 1 : 1=A ( 1) ( 2) 1+0+13 ( 1)2 1+16 ( 1)2 ( 2),A= 12F= 121s+0 1s2+131s 1+161s+2f(t)= 12+13et+16e 2tBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolvef (t)+f(t) 2 t0f(z)dz=t,f(0)=0F=ss2(s 1)(s+2)=As+Bs2+Cs 1+Ds+2s=As(s 1)(s+2)+B(s 1)(s+2)+Cs2(s+2)+Ds2(s 1)s=0 :0=B ( 1) 2,B=0s=1 :1=C 12 3,C=13s= 2 : 2=D ( 2)2 ( 3),D=16s= 1 : 1=A ( 1) ( 2) 1+0+13 ( 1)2 1+16 ( 1)2 ( 2),A= 12F= 121s+0 1s2+131s 1+161s+2f(t)= 12+13et+16e 2tBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolvef (t)+f(t) 2 t0f(z)dz=t,f(0)=0F=ss2(s 1)(s+2)=As+Bs2+Cs 1+Ds+2s=As(s 1)(s+2)+B(s 1)(s+2)+Cs2(s+2)+Ds2(s 1)s=0 :0=B ( 1) 2,B=0s=1 :1=C 12 3,C=13s= 2 : 2=D ( 2)2 ( 3),D=16s= 1.

9 1=A ( 1) ( 2) 1+0+13 ( 1)2 1+16 ( 1)2 ( 2),A= 12F= 121s+0 1s2+131s 1+161s+2f(t)= 12+13et+16e 2tBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolvef (t)+f(t) 2 t0f(z)dz=t,f(0)=0F=ss2(s 1)(s+2)=As+Bs2+Cs 1+Ds+2s=As(s 1)(s+2)+B(s 1)(s+2)+Cs2(s+2)+Ds2(s 1)s=0 :0=B ( 1) 2,B=0s=1 :1=C 12 3,C=13s= 2 : 2=D ( 2)2 ( 3),D=16s= 1 : 1=A ( 1) ( 2) 1+0+13 ( 1)2 1+16 ( 1)2 ( 2),A= 12F= 121s+0 1s2+131s 1+161s+2f(t)= 12+13et+16e 2tBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolvef (t)+f(t) 2 t0f(z)dz=t,f(0)=0F=ss2(s 1)(s+2)=As+Bs2+Cs 1+Ds+2s=As(s 1)(s+2)+B(s 1)(s+2)+Cs2(s+2)+Ds2(s 1)s=0 :0=B ( 1) 2,B=0s=1 :1=C 12 3,C=13s= 2 : 2=D ( 2)2 ( 3),D=16s= 1.

10 1=A ( 1) ( 2) 1+0+13 ( 1)2 1+16 ( 1)2 ( 2),A= 12F= 121s+0 1s2+131s 1+161s+2f(t)= 12+13et+16e 2tBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolvef (t)+f(t) 2 t0f(z)dz=t,f(0)=0F=ss2(s 1)(s+2)=As+Bs2+Cs 1+Ds+2s=As(s 1)(s+2)+B(s 1)(s+2)+Cs2(s+2)+Ds2(s 1)s=0 :0=B ( 1) 2,B=0s=1 :1=C 12 3,C=13s= 2 : 2=D ( 2)2 ( 3),D=16s= 1 : 1=A ( 1) ( 2) 1+0+13 ( 1)2 1+16 ( 1)2 ( 2),A= 12F= 121s+0 1s2+131s 1+161s+2f(t)= 12+13et+16e 2tBernd Schr oderLouisiana Tech University, College of Engineering and ScienceLaplace Transforms and Integral Equationslogo1 Transforms and New FormulasAn ExampleDouble CheckSolvef (t)+f(t) 2 t0f(z)dz=t,f(0)=0F=ss2(s 1)(s+2)=As+Bs2+Cs 1+Ds+2s=As(s 1)(s+2)+B(s 1)(s+2)+Cs2(s+2)+Ds2(s 1)s=0 :0=B ( 1) 2,B=0s=1 :1=C 12 3,C=13s= 2 : 2=D ( 2)2 ( 3),D=16s= 1.