How can I solve this differential equation? : # xy \ dx-(x^2+1) \ dy = 0 #

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Answer 1 · 2018-02-14T16:26:18

# y = Asqrt(x^2+1) #

we have in differential form:

# xy \ dx-(x^2+1) \ dy = 0 #

If we put in standard form, and collect terms:

# 1/yy \ dy/dx = x/(x^2+1) #

Which is a First Order Separable Ordinary Differential Equation, so we can separate the variables to get:

# int \ 1/y \ dy = int \ x/(x^2+1) \ dx #

We can manipulate the RHS integral as follows:

# int \ 1/y \ dy = 1/2 \ int \ (2x)/(x^2+1) \ dx #

And now both integrals are standard results, so integrating give us:

# ln|y| = 1/2ln|x^2+1| + C #

Noting that we require areal solution, and writing #C=lnA#, we get:

# ln y = ln Asqrt(x^2+1) #

Giving us the General Solution:

# y = Asqrt(x^2+1) #