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The substitution V = In u can transform the differential equation (DE) dy/dx + y = 2(y In y) %3D %3D into the DE dv/dx +1 2v. %3D O True False

The substitution V = In u can transform the differential equation (DE) dy/dx + y = 2(y In y) %3D %3D into the DE dv/dx +1 2v. %3D O True False

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The substitution v= In y can transform the differential equation (DE) dy/dx+y= 2(yln y) %3D into the DE dv/dx +1= 2v. O True False
Transcribed Image Text:The substitution v= In y can transform the differential equation (DE) dy/dx+y= 2(yln y) %3D into the DE dv/dx +1= 2v. O True False
Expert Solution
Step 1

We have dydx+y=2ylny                    ...(1)

Let v=lny

Taking e on both sides:

ev=elny⇒ev=y⇒y=ev

Hence, we get y=ev

Now, differentiate v=lny with respect to x:

dvdx=ddxlny⇒dvdx=1ydydx⇒dydx=ydvdx

Hence, we got v=lny or y=ev and dydx=ydvdx

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