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# Differential Equations/differntial equation question

Question
(x^2)y''+xy'+y=x  with y(1)=y'(1)=1. Then what is the value of y(e^π/2)

First solve the associated homogeneous problem: (x^2)y''+xy'+y=0, by letting y=x^n
Substituting that into the DE gives: (x^n)*(n^2+1)=0 ==> n=-i or +i.
Thus, y1=x^i or y2=x^(-i).  Using Euler's formula e^(bi)=cos(b)+isin(b), we get
y1=cos(ln(x))+isin(ln(x)), and y2=cos(ln(x))-isin(ln(x)).  Combining them as a
linear combination to obtain y(x), gives y(x)=k1*y1+k2*y2= (k1+k2)cos(ln(x))+(k1-k2)isin(ln(x))
Letting C1=k1+k2 and C2=(k1-k2)i, gives the homogeneous soluiton: y(x)=C1*cos(ln(x))+C2*sin(ln(x)).

Now for the particular solution to (x^2)y''+xy'+y=x
Solve by Variation of Parameters, to obtain y(x)=(1/2)x

Thus, the complete solution is y(x)=C1*cos(ln(x))+C2*sin(ln(x))+(1/2)x.

Now you can substitute the IC's in to C1=1/2 and C2=1/2....then let x=e^n/2...

OK, good?
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Differential Equations

Volunteer

#### Abe Mantell

##### Expertise

Hello, I am a college professor of mathematics and regularly teach all levels from elementary mathematics through differential equations, and would be happy to assist anyone with such questions!

##### Experience

Over 15 years teaching at the college level.

Organizations
NCTM, NYSMATYC, AMATYC, MAA, NYSUT, AFT.

Education/Credentials
B.S. in Mathematics from Rensselaer Polytechnic Institute
M.S. (and A.B.D.) in Applied Mathematics from SUNY @ Stony Brook