Calculus of a Single Variable, Chapter 6, 6.2, Section 6.2, Problem 5

The general solution of a differential equation in a form of y'=f(x,y) can be evaluated using direct integration. The derivative of y denoted as y' can be written as (dy)/(dx) then y'= f(x) can be expressed as (dy)/(dx)= f(x) .
For the problem: y'=5x/y , we let y'=(dy)/(dx) to set it up as:
(dy)/(dx)= 5x/y
Cross-multiply dx to the right side:
(dy)= 5x/ydx
Cross-multiply y to the left side:
ydy=5xdx
Apply direct integration on both sides:
int ydy=int 5xdx
Apply basic integration property: int c*f(x)dx = c int f(x) dx on the right side.
int ydy=int 5xdx
int ydy=5int xdx
Apply Power Rule for integration: int u^n du= u^(n+1)/(n+1)+C on both sides.
For the left side, we get:
int y dy = y^(1+1)/(1+1)
= y^2/2
For the right side, we get:
int x dx = x^(1+1)/(1+1)+C
= x^2/2+C
Note: Just include the constant of integration "C" on one side as the arbitrary constant of a differential equation.
Combining the results from both sides, we get the general solution of the differential equation as:
y^2/2=x^2/2+C
or y =+-sqrt(x^2/2+C)