60.1.430 problem 432

Internal problem ID [10444]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 432
Date solved : Monday, January 27, 2025 at 07:44:00 PM
CAS classification : [_rational]

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

✓ Solution by Maple

Time used: 0.053 (sec). Leaf size: 78

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

\[ y-\operatorname {RootOf}\left (-a \,\operatorname {arcsinh}\left (\frac {\operatorname {RootOf}\left (-2 a y+a^{2}+x^{2}+2 a \textit {\_Z} +\textit {\_Z}^{2}\right )}{x}\right )-x \sqrt {\frac {a \left (-2 \operatorname {RootOf}\left (-2 a y+a^{2}+x^{2}+2 a \textit {\_Z} +\textit {\_Z}^{2}\right )+2 \textit {\_Z} -a \right )}{x^{2}}}+c_{1} \right ) = 0 \]

✓ Solution by Mathematica

Time used: 0.740 (sec). Leaf size: 70

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

\[ \text {Solve}\left [\left \{y(x)=\frac {2 a x K[1]+x^2 K[1]^2+a^2+x^2}{2 a},x=-\frac {a \text {arcsinh}(K[1])}{\sqrt {K[1]^2+1}}+\frac {c_1}{\sqrt {K[1]^2+1}}\right \},\{y(x),K[1]\}\right ] \]