60.1.490 problem 493

Internal problem ID [10504]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 493
Date solved : Monday, January 27, 2025 at 08:24:23 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} \left (y^{2}-2 a x +a^{2}\right ) {y^{\prime }}^{2}+2 a y y^{\prime }+y^{2}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.339 (sec). Leaf size: 113

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

\begin{align*} y &= 0 \\ \left [x \left (\textit {\_T} \right ) &= \frac {\sqrt {\textit {\_T}^{2}+1}\, {\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )}^{2} a^{2}+\left (-2 a c_{1} \sqrt {\textit {\_T}^{2}+1}-2 a^{2}\right ) \operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )+\left (a^{2}+c_{1}^{2}\right ) \sqrt {\textit {\_T}^{2}+1}+2 c_{1} a}{2 \sqrt {\textit {\_T}^{2}+1}\, a}, y \left (\textit {\_T} \right ) = \frac {\left (-a \,\operatorname {arctanh}\left (\frac {1}{\sqrt {\textit {\_T}^{2}+1}}\right )+c_{1} \right ) \textit {\_T}}{\sqrt {\textit {\_T}^{2}+1}}\right ] \\ \end{align*}

✓ Solution by Mathematica

Time used: 2.366 (sec). Leaf size: 149

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

\[ \text {Solve}\left [\left \{x=\frac {a^2 K[1]^2+2 a K[1] y(K[1])+K[1]^2 y(K[1])^2+y(K[1])^2}{2 a K[1]^2},y(x)=\exp \left (\int _1^{K[1]}\frac {1}{K[2] \left (K[2]^2+1\right )}dK[2]\right ) \int _1^{K[1]}\frac {a \exp \left (-\int _1^{K[3]}\frac {1}{K[2] \left (K[2]^2+1\right )}dK[2]\right )}{K[3]^2+1}dK[3]+c_1 \exp \left (\int _1^{K[1]}\frac {1}{K[2] \left (K[2]^2+1\right )}dK[2]\right )\right \},\{y(x),K[1]\}\right ] \]