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.