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

##### 1 Answer

Feb 14, 2018

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

#### Explanation:

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

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

Giving us the General Solution:

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