81.2.2 problem 2

Internal problem ID [18610]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter II. Change of variable. Exercises at page 20
Problem number : 2
Date solved : Tuesday, January 28, 2025 at 12:04:02 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} y^{\prime \prime }-{y^{\prime }}^{2}-y {y^{\prime }}^{3}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 30

dsolve(diff(y(x),x$2)-diff(y(x),x)^2-y(x)*diff(y(x),x)^3=0,y(x), singsol=all)
 

\begin{align*} y \left (x \right ) &= c_{1} \\ y \left (x \right )-{\mathrm e}^{-y \left (x \right )} c_{1} -\frac {y \left (x \right )^{2}}{2}-x -c_{2} &= 0 \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.565 (sec). Leaf size: 96

DSolve[D[y[x],{x,2}]-D[y[x],x]^2-y[x]*D[y[x],x]^3==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1}^2}{2}+\text {$\#$1}-e^{-\text {$\#$1}} c_1\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1}^2}{2}+\text {$\#$1}-e^{-\text {$\#$1}} (-c_1)\&\right ][x+c_2] \\ y(x)\to \text {InverseFunction}\left [-\frac {\text {$\#$1}^2}{2}+\text {$\#$1}-e^{-\text {$\#$1}} c_1\&\right ][x+c_2] \\ \end{align*}