60.7.81 problem 1672 (book 6.81)

Internal problem ID [11670]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1672 (book 6.81)
Date solved : Monday, January 27, 2025 at 11:29:48 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Solution by Maple

Time used: 0.075 (sec). Leaf size: 41

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

\begin{align*} y &= \frac {c_{2} c_{1} +2 \sqrt {c_{1} x -1}}{c_{1}} \\ y &= \frac {c_{2} c_{1} -2 \sqrt {c_{1} x -1}}{c_{1}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.837 (sec). Leaf size: 48

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

\[ y(x)\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (K[1]^2+1\right )}dK[1]\&\right ]\left [c_1-\frac {1}{2} \log (K[2])\right ]dK[2]+c_2 \]