Solve the differential equation dy/ dx = (1 + y^2) * e^x .

To solve the given expression, we'll re-write
it:

dy = [(1 + y^2) *
e^x]*dx

Now, we'll have to integrate both sides, to
calculate y:

Int dy = Int [(1 + y^2) *
e^x]*dx

We'll integrate with respect to x, to the right
side of the equation, so the sum 1 + y^2 will be considered as a
constant:

y = (1 + y^2)*Int e^x
dx

We'll divide by (1 +
y^2):

y/(1 + y^2) = e^x + C

y
= e^x + y^2*e^x + C + Cy^2

y^2(C + e^x) - y + e^x + C=
0

The equation has solutions if it' discriminant is
positive or zero:

delta = b^2 -
4ac

a = C + e^x

b =
-1

c = e^x + C

delta = 1 -
4(e^x + C)^2

delta>0

1
- 4(e^x + C)^2>0

4(e^x + C)^2 <
1

(e^x + C)^2 < 1/4

e^x
+ C < +/- 1/2

e^x < C +
1/2

ln e^x < ln(C+/-
1/2)

x < ln (C+/-
1/2)