Question

The solution of differential equation $\dfrac{{dy}}{{dx}} = 2x$ is

Answer 1

Hint: We are provided a differential equation in the given question. As a result, we must use integration methods to solve the given differential equation. We will categorise the differential equation and then answer it based on the type of equation. Then, after computing an indefinite integral, we will add an arbitrary constant c that symbolises a family of curves.

Complete answer:
We have given $\dfrac{{dy}}{{dx}} = 2x$
$ \Rightarrow \dfrac{{dy}}{{dx}} = 2x$
We will multiply dx on both sides
$ \Rightarrow \dfrac{{dy}}{{dx}}dx = 2xdx$
$ \Rightarrow dy = 2xdx$
We will integrate both sides with respect to dx.
$ \Rightarrow \int {dy} = \int {2xdx} $
We know that by power rule of integration \[\smallint {x^n}dx = \dfrac{{{x^{n + 1}}}}{{n + 1}}\]
$ \Rightarrow y = 2\dfrac{{{x^2}}}{2} + C$
$ \Rightarrow y = {x^2} + C$
Hence, the integration of 2x is ${x^2} + C$ where C is constant.

Note:
The indefinite integrals of certain functions may have more than one answer in different forms. All of these forms, however, are correct and interchangeable. We get the family of curves from the indefinite integral since we don't know the exact value of the constant. Only if we are provided a point on the function can we calculate the exact value of the arbitrary constant.