Question

How do you find all the solutions of the differential equation\[\dfrac{dy}{dx}={{x}^{2}}{{y}^{2}}+{{x}^{2}}\] ?

Answer 1

Hint: We can see that the equation is in variable separable form, it is one method to solve the differential equation. Thus, by separating the variables moving y variable, which includes dy to one side and x variable, which includes dx to other and then applying integration on both sides and thus integrating using the general rule of integration.

Complete step-by-step answer:
This question belongs to the concept of methods of solving differential equations, which means finding the solution of differential equations.
As we have four ways of solving a differential equation by Separation of Variables equations, by First Order Linear equations, by Homogeneous equations, by Bernoulli method.
Here we can see that the equation is in variable separable form thus by this method we are going to solve the question.
First of all, let us get a brief introduction of what is a variable separable method of solving differential equations.
In variable separable methods we transform the given ordinary or partial differential equation in such a way that each of the two variables are on the opposite side of the differential equation.
The equation given in the question is
\[\dfrac{dy}{dx}={{x}^{2}}{{y}^{2}}+{{x}^{2}}\]
\[\Rightarrow \dfrac{dy}{dx}={{x}^{2}}({{y}^{2}}+1)\]
Since it is in variable separable form, we can separate the variables to make the integration easy.
After separating the variable, we get the below equation
\[\Rightarrow \dfrac{dy}{{{y}^{2}}+1}={{x}^{2}}dx\]
Now, we can see that this equation can be integrated directly, so we will integrate on both sides.
\[\Rightarrow \int{\dfrac{dy}{{{y}^{2}}+1}}=\int{{{x}^{2}}}dx\]
As we know
\[\Rightarrow \int{\dfrac{dy}{{{y}^{2}}+1}}={{\tan }^{-1}}y\]
Therefore, the equation after integration becomes
\[\Rightarrow {{\tan }^{-1}}y=\dfrac{{{x}^{3}}}{3}+C\]
Since we are integrating, it is mandatory to add a constant after integration as integration gives us an extra term.
Hence the solution for the differential equation is
\[{{\tan }^{-1}}y=\dfrac{{{x}^{3}}}{3}+C\]

Note: Since there are many ways to find the solution of a differential equation for different types of differential equation, it is important to find the correct form and method to solve (in this question it is a variable separable method). Also remember to add the constant with the equation after integration. Take note of the integration formula used in the above question for future use.