Example 3B \begin {align} y^{\prime \prime } & =\sqrt {1+\left ( y^{\prime }\right ) ^{2}}\tag {1}\\ y\left ( 0\right ) & =1\nonumber \end {align} This is slightly alternative way to solving the ode. Let \(p=y^{\prime }\) then \(y^{\prime \prime }=p^{\prime }\). Hence the ode becomes\begin {equation} p^{\prime }=\sqrt {1+p^{2}} \tag {2} \end {equation} Solving this as first order gives\[ p\left ( x\right ) =\sinh \left ( x+c_{1}\right ) \] But \(p=y^{\prime }\) hence the above becomes\[ y^{\prime }\left ( x\right ) =\sinh \left ( x+c_{1}\right ) \] Integrating gives\begin {align} y & =\int \sinh \left ( x+c_{1}\right ) dx+c_{2}\nonumber \\ & =\cosh \left ( x+c_{1}\right ) +c_{2} \tag {3} \end {align}

Now we need to apply IC’s to find \(c_{1},c_{2}.\) We only have one IC \(y\left ( 0\right ) =1\). Applying this to the above solution gives\begin {align*} 1 & =\cosh \left ( c_{1}\right ) +c_{2}\\ c_{2} & =1-\cosh \left ( c_{1}\right ) \end {align*}

Hence (3) becomes\[ y\left ( x\right ) =\cosh \left ( x+c_{1}\right ) +1-\cosh \left ( c_{1}\right ) \]