Question

How do you find all solutions of the differential equation $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=0$ ?

Answer 1

Hint : We first explain the terms $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ and $\dfrac{dy}{dx}$ where $y=f\left( x \right)$ . We then need to integrate the equation twice to find all the solutions of the differential equation $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=0$ . We take two constant terms for the integration. We get the equation of a line.

Complete step-by-step answer :
We have given a differential equation $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=0$ .
Here $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}$ defines the second order differentiation which is expressed as $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=\dfrac{d}{dx}\left( \dfrac{dy}{dx} \right)$ .
Again $\dfrac{dy}{dx}$ defines the first order differentiation which is expressed as $\dfrac{dy}{dx}=\dfrac{d}{dx}\left( y \right)$.
The main function is $y=f\left( x \right)$ .
We have to find the antiderivative or the integral form of the equation.
We know the differentiation of constant terms is 0 which gives the anti-derivative or the integral form of 0 is constant term.
So, $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=\dfrac{d}{dx}\left( \dfrac{dy}{dx} \right)=0$ .
Integrating both sides we get $\int{d\left( \dfrac{dy}{dx} \right)}=b$ . Here the term $b$ is a constant.
We get $\left( \dfrac{dy}{dx} \right)=b$ .
We again need to integrate the function $\left( \dfrac{dy}{dx} \right)=b$ to find the solution of the differential equation $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=0$ . We get $\int{\dfrac{dy}{dx}}=\int{b}$ .
Simplifying the differential form, we get $\int{dy}=b\int{dx}+c$ .
Here $c$ is another constant.
The equation becomes $y=bx+c$ .
All the solutions of the differential equation $\dfrac{{{d}^{2}}y}{d{{x}^{2}}}=0$ is $y=bx+c$ .
So, the correct answer is “ $y=bx+c$ ”.

Note : The solution of the differential equation is the equation of a line. The second order differentiation of $y=bx+c$ gives the rate of change of the slope of the line. It is always equal to the value of 0.