15.18.22 problem 22

Internal problem ID [3265]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 22
Date solved : Monday, January 27, 2025 at 07:29:43 AM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 59

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

\begin{align*} y \left (x \right ) &= -\frac {c_{1} \left ({\mathrm e}^{\frac {c_{2} +x}{c_{1}}}-{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\ y \left (x \right ) &= \frac {c_{1} \left ({\mathrm e}^{\frac {c_{2} +x}{c_{1}}}-{\mathrm e}^{\frac {-x -c_{2}}{c_{1}}}\right )}{2} \\ \end{align*}

✓ Solution by Mathematica

Time used: 13.494 (sec). Leaf size: 84

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

\begin{align*} y(x)\to -\frac {\sinh \left (\sqrt {e^{2 c_1}} (x+c_2)\right )}{\sqrt {e^{2 c_1}}} \\ y(x)\to \frac {\sinh \left (\sqrt {e^{2 c_1}} (x+c_2)\right )}{\sqrt {e^{2 c_1}}} \\ y(x)\to -x-c_2 \\ y(x)\to x+c_2 \\ \end{align*}