29.27.8 problem 774

Internal problem ID [5358]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Various 27
Problem number : 774
Date solved : Monday, January 27, 2025 at 11:17:14 AM
CAS classification : [_quadrature]

\begin{align*} {y^{\prime }}^{2}-2 y^{\prime }-y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.037 (sec). Leaf size: 66

dsolve(diff(y(x),x)^2-2*diff(y(x),x)-y(x)^2 = 0,y(x), singsol=all)
 

\begin{align*} \frac {-\sqrt {y \left (x \right )^{2}+1}+\operatorname {arcsinh}\left (y \left (x \right )\right ) y \left (x \right )-1+\left (-c_{1} +x \right ) y \left (x \right )}{y \left (x \right )} &= 0 \\ \frac {\sqrt {y \left (x \right )^{2}+1}-\operatorname {arcsinh}\left (y \left (x \right )\right ) y \left (x \right )-1+\left (-c_{1} +x \right ) y \left (x \right )}{y \left (x \right )} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.964 (sec). Leaf size: 104

DSolve[(D[y[x],x])^2-2*D[y[x],x]-y[x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\text {$\#$1}^2+1}+\text {$\#$1} \log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )+1}{\text {$\#$1}}\&\right ][-x+c_1] \\ y(x)\to \text {InverseFunction}\left [-\frac {\sqrt {\text {$\#$1}^2+1}}{\text {$\#$1}}-\log \left (\sqrt {\text {$\#$1}^2+1}-\text {$\#$1}\right )+\frac {1}{\text {$\#$1}}\&\right ][x+c_1] \\ y(x)\to 0 \\ \end{align*}