Understanding Differential Equations and their Solutions

[beetle2]

Homework Statement

Hi Guys,
I've got a couple of examples of de's in my lecture notes. one is:
y'=1+x^2
which has a general solution of y= x+\frac{x^3}{3} + c
which i I understand they have taken the integral wrt x


the second is

y'=1+y^2 which has a general solution of y=tan(x+c)

Can some one please explain to me how that got the second solution?

 

[Mark44]

Homework Statement

Hi Guys,
I've got a couple of examples of de's in my lecture notes. one is:
y'=1+x^2
which has a general solution of y= x+\frac{x^3}{3} + c
which i I understand they have taken the integral wrt x


the second is

y'=1+y^2 which has a general solution of y=tan(x+c)

Can some one please explain to me how that got the second solution?

Both problems can be done by the technique of separation of variables.

For the first, you have
dy/dx = 1 + x^2
==> dy = (1 + x^2)dx
Integrate both sides to get what you already have shown.

For the second, you have
dy/dx = 1 + y^2
==> dy/(1 + y^2) = dx
Now integrate both sides to get
arctan(y) = x + C ==> y = tan(x + C).

 

[beetle2]

Thanks mate I had forgoten about separation of variables.
I was integrating both sides wrt x in the second example.

 

[Mark44]

Thanks mate I had forgoten about separation of variables.
I was integrating both sides wrt x in the second example.

And good luck with that!