83.22.19 problem 19

Internal problem ID [19273]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter IV. Equations of the first order but not of the first degree. Exercise IV (E) at page 63
Problem number : 19
Date solved : Tuesday, January 28, 2025 at 01:19:11 PM
CAS classification : [_quadrature]

\begin{align*} y&=a y^{\prime }+\sqrt {{y^{\prime }}^{2}+1} \end{align*}

✓ Solution by Maple

Time used: 0.044 (sec). Leaf size: 112

dsolve(y(x)=a*diff(y(x),x)+sqrt(1+diff(y(x),x)^2),y(x), singsol=all)
 

\begin{align*} \left (\int _{}^{y \left (x \right )}\frac {1}{-\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right ) a^{2}-\int _{}^{y \left (x \right )}\frac {1}{-\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} -c_{1} +x &= 0 \\ -\left (\int _{}^{y \left (x \right )}\frac {1}{\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} \right ) a^{2}+\int _{}^{y \left (x \right )}\frac {1}{\textit {\_a} a +\sqrt {\textit {\_a}^{2}+a^{2}-1}}d \textit {\_a} -c_{1} +x &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.972 (sec). Leaf size: 210

DSolve[y[x]==a*D[y[x],x]+Sqrt[1+D[y[x],x]^2],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\frac {a \left (\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a+1\right )+\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}+a-1\right )\right )-(a+1) \log \left ((a-1) \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}\right )\right )}{a^2-1}\&\right ]\left [\frac {x}{a^2-1}+c_1\right ] \\ y(x)\to \text {InverseFunction}\left [\frac {a \left (\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}-a-1\right )+\log \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}+a+1\right )\right )-(a-1) \log \left ((a+1) \left (\sqrt {\text {$\#$1}^2+a^2-1}-\text {$\#$1}\right )\right )}{a^2-1}\&\right ]\left [\frac {x}{a^2-1}+c_1\right ] \\ y(x)\to 1 \\ \end{align*}