An ordinary differential equation (ODE) has differential equation for a function with single variable. A first order ODE follows `(dy)/(dx)= f(x,y)` .

It can also be in a form of `N(y) dy= M(x) dx` as **variable separable
differential equation..**

To be able to set-up the problem as `N(y) dy= M(x) dx` , we let` y' = (dy)/(dx).`

The problem: `y'=sqrt(x)y` becomes:

`(dy)/(dx)=sqrt(x)y`

Rearrange by cross-multiplication, we get:

`(dy)/y=sqrt(x)dx`

Apply direct integration on both sides:` int (dy)/y=int sqrt(x)dx` to solve for the general solution of a differential equation.

For the left side, we applying **basic integration formula for
logarithm**:

`int(dy)/y= ln|y|`

For the right side, we apply the Law of Exponent: `sqrt(x)=x^(1/2)` then
follow the **Power Rule of integration**: `int x^n dx =
x^(n+1)/(n+1)+C`

`int sqrt(x)* dx= int x^(1/2)* dx`

`= x^(1/2+1)/(1/2+1)+C`

` = x^(3/2)/(3/2)+C`

` = x^(3/2)*(2/3)+C`

` = (2x^(3/2))/3+C`

Combining the results from both sides, we get the **general solution
of the differential equation** as:

`ln|y|=(2x^(3/2))/3+C`

or

`y =e^(((2x^(3/2))/3+C))`

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