60.7.42 problem 1632 (6.42)

Internal problem ID [11631]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1632 (6.42)
Date solved : Monday, January 27, 2025 at 11:27:55 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], _Lagerstrom, [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Solution by Maple

Time used: 0.141 (sec). Leaf size: 23

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

\[ y = \frac {\tan \left (\left (x +c_{2} \right ) \sqrt {c_{1} a}\right ) \sqrt {c_{1} a}}{a} \]

✓ Solution by Mathematica

Time used: 0.161 (sec). Leaf size: 90

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

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