60.1.464 problem 467

Internal problem ID [10478]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 467
Date solved : Monday, January 27, 2025 at 07:54:50 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} y {y^{\prime }}^{2}-4 x y^{\prime }+y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.166 (sec). Leaf size: 92

dsolve(y(x)*diff(y(x),x)^2-4*x*diff(y(x),x)+y(x) = 0,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )-\int _{}^{\textit {\_Z}}\frac {\textit {\_a}^{2}+\sqrt {-\textit {\_a}^{2}+4}-2}{\textit {\_a} \left (\textit {\_a}^{2}-3\right )}d \textit {\_a} +c_{1} \right ) x \\ y &= \operatorname {RootOf}\left (-\ln \left (x \right )+\int _{}^{\textit {\_Z}}-\frac {\textit {\_a}^{2}-\sqrt {-\textit {\_a}^{2}+4}-2}{\textit {\_a} \left (\textit {\_a}^{2}-3\right )}d \textit {\_a} +c_{1} \right ) x \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.615 (sec). Leaf size: 65

DSolve[y[x] - 4*x*D[y[x],x] + y[x]*D[y[x],x]^2==0,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \left (y(x)\text {/.}\, \left \{4 x&=\text {K$\$$15315806} y(x)+\frac {y(x)}{\text {K$\$$15315806}},c_1 \exp \left (\int _1^{\text {K$\$$15315806}}\frac {1-K[1]^2}{K[1] \left (K[1]^2-3\right )}dK[1]\right )=y(x)\right \}\right ) \\ y(x)\to 0 \\ \end{align*}