Question

Discuss how the method of reduction of order considered in this section can be applied to the third order differential equation$\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{=}\sqrt{\mathbf{1}\mathbf{+}{\mathbf{(}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{)}}^{\mathbf{2}}}$. Carry out your ideas and solve the equation.

Short answer

The solution of the above question is $y=\sinh (x+c)+kx+h$.

Step-by-step solution

Method of reduction order

The method of reduction of order to solve a second order differential equation is based on the idea of solving first order differential equations one after the other which have been derived from the original second order equation to simplify the problem.

Method of reduction order will be applied:

In this question we will apply the concept Method of reduction order;

$$\begin{gathered} y\text{'}\text{'}\text{'}=\sqrt{1+{(y\text{'}\text{'})}^2} \\ y\text{'}\text{'}\text{'}=s\text{'} \\ s\text{'}=\sqrt{1+s^2} \\ \frac{ds}{dx}=\sqrt{1+s^2} \\ \int \frac{1}{\sqrt{1+s^2}}ds=\int dx \\ x+c=\int \frac{1}{\sqrt{1+s^2}}ds \\ x+c={\sinh }^{-1}s \\ s=\sinh (x+c) \\ y\text{'}\text{'}=\sinh (x+c) \\ y\text{'}=\cosh (x+c)+k \\ y=\sinh (x+c)+kx+h \end{gathered}$$

Hence, the final answer is $y=\sinh (x+c)+kx+h$.