Use Laplace transform to solve the following initial-value problemy"+2y′+y=0y(0)=1,y′(0)=1a) e−t+te−tb) et+2tetc) e−t+2tetd) e−t+2te−te) 2e−t+2te−tf) Non of the above

Name: Use Laplace transform to solve the following initial-value problemy&quot;+2y′+y=0y(0)=1,y′(0)=1a) e−t+te−tb) et+2tetc) e−t+2tetd) e−t+2te−te) 2e−t+2te−tf) Non of the above
Uploaded: 2022-02-23T17:22:57+00:00
Description: Use Laplace transform to solve the following initial-value problemy&quot;+2y′+y=0y(0)=1,y′(0)=1a) e−t+te−tb) et+2tetc) e−t+2tetd) e−t+2te−te) 2e−t+2te−tf) Non of the above

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Use Laplace transform to solve the following initial-value problemy&quot;+2y′+y=0y(0)=1,y′(0)=1a) e−t+te−tb) et+2tetc) e−t+2tetd) e−t+2te−te) 2e−t+2te−tf) Non of the above

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Step 1Given initial value problem,y″+2y′+y=0y(0)=1y′(0)=1Step 2Taking inverse Laplace transform,L[y″+2y′+y]=0L[y″]+2L[y′]+L[y]=0Use the formula such that,L[y″]=s2L[y]−sy(0)−y′(0)L[y′]=sL[y]−y(0)Then,s2L[y]−sy(0)−y′(0)+2[sL[y]−y(0)]+L[y]=0s2L[y]−s−1+2[sL[y]−1]+L[y]=0s2L[y]−s−1+2sL[y]−2+L[y]=0(s2+2s+1)L[y]−s−3=0L[y]=s+3s2+2s+1Step 3Taking inverse Laplace transform of both sides,y=L−1[s+3s2+2s+1]=L−1[s+1(s+1)2+2(s+1)2]=L−1[1s+1]+2L−1[1(s+1)2]=e−t+2te−tStep 4Hence, the solution of given initial value problem isy(t)=e−t+2te−t

Use Laplace transform to solve the following initial-value problemy"+2y′+y=0y(0)=1,y′(0)=1a) e−t+te−tb) et+2tetc) e−t+2tetd) e−t+2te−te) 2e−t+2te−tf) Non...

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