Calculus of a Single Variable, Chapter 6, 6.1, Section 6.1, Problem 51

An ordinary differential equation (ODE) is differential equation for the derivative of a function of one variable. When an ODE is in a form of y'=f(x,y) , this is just a first order ordinary differential equation.
We may express y' as (dy)/(dx) to write in a form of (dy)/(dx)= f(x,y) and apply variable separable differential equation: N(y)dy = M(x) dx .
The given problem: (dy)/(dx) = xe^(x^2) can be rearrange as:
dy= xe^(x^2) dx
Apply direct integration on both sides:
int dy= int xe^(x^2) dx
For the left side, we apply basic integration property: int (dy)=y .
For the right side, we may apply u-substitution by letting: u = x^2 then du =2x dx or (du)/2=x dx .
The integral becomes:
int xe^(x^2) dx=int e^(x^2) *xdx
=int e^(u) *(du)/2
Apply the basic integration property: int c*f(x)dx= c int f(x) dx .
int e^(u) *(du)/2=(1/2)int e^(u) du
Apply basic integration formula for exponential function:
(1/2)int e^(u) du=(1/2)e^(u)+C
Plug-in u=x^2 on (1/2)e^(u)+C , we get:
int xe^(x^2) dx=(1/2)e^(x^2)+C
Combining the results from both sides, we get the general solution of differential equation as:
y=(1/2)e^(x^2)+C
or
y=e^(x^2)/2+C