Solve the given differential equation by undetermined coefficientsy″−14′+49y=35x+4

Margie Marx

Margie Marx

Answered

2021-12-19

Solve the given differential equation by undetermined coefficients
y14+49y=35x+4

Answer & Explanation

Ana Robertson

Ana Robertson

Expert

2021-12-20Added 26 answers

Consider the differential equation
y 14+49y=35x+4...(1)
Rewrite the equation,(D214D+49)y=30 where D=d dx 
Auxiliary equation is,f(m)=0
m2-14m+49=0
(m-7)2=0
m=7,7
Hence the complementary solution is,
yc=c1e7x+c2e7x

otoplilp1

otoplilp1

Expert

2021-12-21Added 41 answers

Is it all answer?
RizerMix

RizerMix

Expert

2021-12-29Added 437 answers

no, here is the continuation of the solution
Apply the undetemined coefficient method to find the particular solution as follows:
yp=Ax+B
Find the first and second derivatives,
yp'=A,
yp''=0
Substitute these values in equation (1)
0-14A+49(Ax+B)=35x+4
-14A+49Ax+49B=35x+4
(-14A+49B)+49Ax=35x+4
Compare the coefficients of constant and x terms,
49A=35
-14A+49B=4
Solve the above two equations,
A=57 and B=27 

(A=(35)/(49)),(-14(5/7)+49B=4),(-10+49B=4),(49B=14),(B=2/7):
Hence the particular solution is,
yp=57x+27
The general solution is,
y=yc+yp
yc=c1e7x+c2e7x+57x+27
Therefore, the required general solution is yc=c1e7x+c2e7x+57x+27

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