Chapter D2.1, Problem 68E

a.

To determine

To show: The first order equation of the given differential equation is $u^{\prime }=\frac{\sqrt{1+u^2}}{sx},u\left(1\right)=0$.

b.

To determine

To show: The general solution $u+\sqrt{1+u^2}=C_1{x}^{{}^{\left(\frac{1}{s}\right)}}$ by solving separable equation.

c.

To determine

To evaluate: The value of $C_1$ in the above solution using $u\left(1\right)=0$ and show that the solution u can be written as $u=\frac{1}{2}\left(x^{\frac{1}{s}}-x^{-\left(\frac{1}{s}\right)}\right)$.

d.

To determine

To find: The general solution of the above equation by consider as $y^{\prime }=u$.

e.

To determine

To find: The value $C_2$ of the above equation using the initial condition $y\left(1\right)=0$.

f.

To determine

To conclude: The path of the master is $y=\frac{sx}{2}\left(\frac{x^{\left(\frac{1}{s}\right)}}{s+1}-\frac{x^{-\left(\frac{1}{s}\right)}}{s-1}\right)+\frac{s}{s^2-1}$.

g.

To determine

To sketch: The graphs of the $y=\frac{sx}{2}\left(\frac{x^{\left(\frac{1}{s}\right)}}{s+1}-\frac{x^{-\left(\frac{1}{s}\right)}}{s-1}\right)+\frac{s}{s^2-1}$ functions when $s=1.1,1.3,1.5\text{ and }2.0$, describe the dependence on s which is observed.