60.7.78 problem 1669 (book 6.78)

Internal problem ID [11667]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1669 (book 6.78)
Date solved : Monday, January 27, 2025 at 11:29:46 PM
CAS classification : [[_2nd_order, _exact, _nonlinear], [_2nd_order, _with_linear_symmetries], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} x y^{\prime \prime }+\left (y-1\right ) y^{\prime }&=0 \end{align*}

✓ Solution by Maple

Time used: 0.184 (sec). Leaf size: 24

dsolve(x*diff(diff(y(x),x),x)+(y(x)-1)*diff(y(x),x)=0,y(x), singsol=all)
 

\[ y = \frac {2 c_{1} +\tanh \left (\frac {\ln \left (x \right )-c_{2}}{2 c_{1}}\right )}{c_{1}} \]

✓ Solution by Mathematica

Time used: 0.149 (sec). Leaf size: 120

DSolve[(-1 + y[x])*D[y[x],x] + x*D[y[x],{x,2}] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2-4 K[1]-2 c_1+4}dK[1]\&\right ]\left [-\frac {\log (x)}{2}+c_2\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2-4 K[1]-2 (-1) c_1+4}dK[1]\&\right ]\left [-\frac {\log (x)}{2}+c_2\right ] \\ y(x)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2-4 K[1]-2 c_1+4}dK[1]\&\right ]\left [-\frac {\log (x)}{2}+c_2\right ] \\ \end{align*}