Find the solution to the differential equationAssume x > 0 and let x ( x...

Laila Murphy

Laila Murphy

Answered

2022-11-18

Find the solution to the differential equation
Assume x > 0 and let
x ( x + 1 ) d u d x = u 2 ,
u ( 1 ) = 4.
I started off by doing some algebra to get:
1 u 2 d u = 1 x 2 + x d x .
I then took the partial fraction of the right side of the equation:
1 u 2 d u = ( 1 x 1 x + 1 ) .
I then took the integral of both sides:
1 u = log x log ( x + 1 ) + C .
From here I don't know what to do because we are solving for u ( x ) and I'm not sure how to get that from 1 u

Answer & Explanation

grizintimbp

grizintimbp

Expert

2022-11-19Added 16 answers

Solving for d u d x we have
d u d x = u 2 x ( x + 1 ) d u d x = u 2 x 2 + x d u d x u 2 = 1 x 2 + x .
Integrate both sides & evaluate the integrals:
1 u = log ( x ) log ( x + 1 ) + C 1 u = 1 log ( x ) log ( x + 1 ) + C 1 .
Now apply the initial condition:
1 C 1 log ( 2 ) = 4 C 1 = 1 4 ( 1 + 4 log ( 2 ) ) .
This gives the result.

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