How do you find the general solution of the differential equation $\dfrac{{dy}}{{dx}} = {x^{\dfrac{3}{2}}}$?

Hint:In this question we need to find the general solution of the given differential equation. Differential equations whose general solution we need to find are first in order and degree, so we can find a solution using separation of variable methods. Knowledge of integration is required in finding solutions to differential equations.

Complete step by step solution:
Let us try to find the general solution of the differential
equation$\dfrac{{dy}}{{dx}} = {x^{\dfrac{3}{2}}}$. This differential equation degree and order is one, we will use separation of variables of method.

In the separation of variables method, we separate variables derivative term and variable term to one side of equality.

We have,
$\dfrac{{dy}}{{dx}} = {x^{\dfrac{3}{2}}}$

For separation of variables we will move $dx$term to R.H.S. After performing separation of variable, we have

$dy = {x^{\dfrac{3}{2}}}dx$

Now, integrating both side of the above equation, we have
$\int {dy = \int {{x^{\dfrac{3}{2}}}dx} } $ $eq(1)$

As we know that integration of $\int {dy = y} $and$\int {{x^n}\,dx\, = \,\dfrac{{{x^{n + 1}}}}{{n + 1}}} $.

Putting values of this knowing integration we get,
$y = \dfrac{{{x^{\dfrac{5}{2}}}}}{{\dfrac{5}{2}}}\, + \,C$ (Here C is
integration constant)

$y = \dfrac{{2{x^{\dfrac{5}{2}}}}}{5}\, + \,C$

So the general equation of the differential equation $\dfrac{{dy}}{{dx}} = {x^{\dfrac{3}{2}}}$ is$y = \dfrac{{2{x^{\dfrac{5}{2}}}}}{5}\, + \,C$.

Note: In this question we are asked to find the general solution of the differential equation. To find the value of the integration question we are required to have a condition which satisfies this equation and such solution is called a particular solution of the differential equation. If you write a general solution of differential equation without integration constant means particular solution of differential equation which is wrong and it’s a common mistake done by students. Differential has many applications in physics and chemistry.