Solve the differential equation xyy'+xyy'=y^2+1?

vballa15ei

vballa15ei

Answered question

2022-09-08

Solve the differential equation x y y + x y y = y 2 + 1 ?

Answer & Explanation

Baluttor7

Baluttor7

Beginner2022-09-09Added 17 answers

This is a First Order separable DE, so we can separate the variables to get;

  y y 2 + 1   d y =   1 2 x   d x

The RHS is trivial and for the LHS we can use a substitution:
Let u = y 2 + 1 d u d y = 2 y , or 1 2 ...   d u = ... y   d y

Substituting we get

  1 2 u   d u =   1 2 x   d x

We can now integrate to get:

1 2 ln | u | = 1 2 ln | x | + A
ln | u | = ln | x | + 2 A
ln | u | = ln | x | + ln B (say)
ln | u | = ln B | x |
u = B x
y 2 + 1 = B x
y 2 = B x - 1
y = ± B x - 1

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