`y' = sqrt(x)y` Solve the differential equation
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:
Rearrange by cross-multiplication, we get:
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:
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^(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: