Solve the ordinary differential equation with initial conditions using Laplace transform.y+4y+13y=20e-tThe above equation's answer is as follows:y=AeBx+eCxsin3t-eDxcos3t

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Solve the ordinary differential equation with initial conditions using Laplace transform.y+4y+13y=20e-tThe above equation&#039;s answer is as follows:y=AeBx+eCxsin3t-eDxcos3t

likvau · Accepted Answer

Step 1The given equation is&amp;nbsp;y″+4y′+13y=20e−tThe initial conditions are&amp;nbsp;y(0)=1,y′(0)=3Taking Laplace transform on both sidesL{y″+4y′+13y}=L{20e−t}Apply formula Laplace transform for the derivativesL{f′(t)}=sL{f(t)}−f(0)L{f″(t)}=s2L{f(t)}−sf(0)−f′(0)L{e−t}=1s+1Step 2Using the formulasL{y&quot;+4y′+13y}=L{20e−t}[s2L{y}−sy(0)−y′(0)]+4[sL{y}−y(0)]+13L{y}=20L(e−t)[s2L{y}−s(1)−3]+4[sL{y}−1]+13L{y}=201s+1s2L{y}−s(1)−3+4sL{y}−4+13L{y}=201s+1s2L{y}−s+4sL{y}+13L{y}−7=201s+1s2L{y}+4sL{y}+13L{y}−s−7=201s+1L{y}{s2+4s+13}−s−7=20s+1Step 3L{y}{s2+4s+13}=20s+1+s+7L{y}=1s2+4s+13(20+s2+7s+s+7s+1)L{y}=1s2+4s+13(s2+8s+27s+1)L{y}=s2+8s+27(s+1)(s2+4s+13)Take inverse LaplaceL−1{L{y}}=L−1{s2+8s+27(s+1)(s2+4s+13)}y=L−1{s2+8s+27(s+1)(s2+4s+13)}…(1)Step 4Making the partial fractionss2+8s+27(s+1)(s2+4s+13)=a(s+1)+bs+c(s2+4s+13)Multiply both sides by&amp;nbsp;(s+1)(s2+4s+13)s2+8s+27(s+1)(s2+4s+13)=a(s+1)+bs+c(s2+4s+13)(s2+8s+27)(s+1)(s2+4s+13)(s+1)(s2+4s+13)=a(s+1)(s2+4s+13)(s+1)+(bs+c)(s+1)(s2+4s+13)(s2+4s+13)(s2+8s+27)=a(s2+4s+13)+(bs+c)(s+1)(s2+8s+27)=a(s2+4s+13)+(bs2+cs+bs+c)(s2+8s+27)=(a+b)s2+(4a+b+c)s+13a+cStep 5Equating both sides to get&amp;nbsp;a+b=14a+b+c=813a+c=27Solve for&amp;nbsp;a,&amp;nbsp;b&amp;nbsp;and&amp;nbsp;c&amp;nbsp;to get the following valuesa=2b=-1c=1Therefore,&amp;nbsp;s2+8s+27(s+1)(s2+4s+13)=2(s+1)+(-1)s+1(s2+4s+13)Equation&amp;nbsp;(1)&amp;nbsp;reduces to&amp;nbsp;L-1s2+8s+27(s+1)(s2+4s+13)=L-12(s+1)+(-1)s+1(s2+4s+13)Step 6L-12(s+1)+(-1)s+1(s2+4s+13)=L-12(s+1)+L-1-s+1(s2+4s+13)…(2)Again Rewrite &amp;nbsp;-s+1(s2+4s+13)=-s-1(s2+4s+4+9)=-s+2((s+2)2+9)+3((s+2)2+9)Equation&amp;nbsp;(2)&amp;nbsp;reduces toL-12(s+1)+(-1)s+1(s2+4s+13)=L-12(s+1)+L-1-s+2(s+2)2+9+L-13(s+2)2+9=L-12(s+1)+L-1-s+2(s+2)2+9+3L-11(s+2)2+9=2e-t-e-2tcos(3t)+313e-2tsin(3t)=2e-t-e-2tcos(3t)+e-2tsin(3t)Step 7comparing with the given equationThe answer is :A=2B=-2C=-2D=-2

Solve the ordinary differential equation with initial conditions using Laplace transform.y"+4y'+13y=20e^(-t) , y_0=1 , y'_0=3

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