Question

What is the solution to the differential equation \[\dfrac{dy}{dx}={{y}^{2}}\]?

Answer 1

Hint: Firstly, we have to check the order of the differential equation given. Then choose the correct procedure of solving the equation. After obtaining the answer, if it contains no products or powers, it is said to be linear or else it is called a separable differential equation.

Complete step-by-step solution:
Let us have a look at what a separable differential equation means. A first order equation is said to be separable if after solving for the derivative, if the right hand side can be factored as a formula of just x times a formula of just y.
Let us start solving \[\dfrac{dy}{dx}={{y}^{2}}\].
Upon checking the order of the equation, we find it to be a First Order Separable Ordinary Differential Equation.
A first order separable ordinary differential equation will have the form \[N\left( y \right)\centerdot {{y}^{'}}=M\left( x \right)\]
Let us substitute \[\dfrac{dy}{dx}\] with \[{{y}^{'}}\].
Now we get, \[{{y}^{'}}\] \[={{y}^{2}}\].
In the next step, we would be rewriting the equation in the form of a first order equation.
The general form of the First Order Separable ODE is \[F\left( x,y \right)=f\left( x \right)\]
Now we are supposed to write our equation in this form.
That would be \[\dfrac{1}{{{y}^{2}}}{y}'\;=1\]
Now on differentiating this equation, we get:
Differentiate it using the \[\dfrac{u}{v}\] formula.
We get, \[\dfrac{-1}{y}=x+{{c}_{1}}\]
On further isolating this equation, we obtain the equation as
\[y=-\dfrac{1}{x+{{c}_{1}}}\]
Hence the solution of the given differential equation is \[y=-\dfrac{1}{x+{{c}_{1}}}\].

Note: We must have a note that differential equations cannot be solved using symbolic analysis. There exist two methods in solving the differential equations: they are- separation of variables and integrating factor. We must be accurate in choosing the method in proceeding with the method.