Internal problem ID [8502]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 922.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _rational]
Solve \begin {gather*} \boxed {y^{\prime }-\frac {y^{2}}{y^{2}+y^{\frac {3}{2}}+\sqrt {y}\, x^{2}-2 y^{\frac {3}{2}} x +y^{\frac {5}{2}}+x^{3}-3 y x^{2}+3 x y^{2}-y^{3}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 47
dsolve(diff(y(x),x) = y(x)^2/(y(x)^2+y(x)^(3/2)+y(x)^(1/2)*x^2-2*y(x)^(3/2)*x+y(x)^(5/2)+x^3-3*x^2*y(x)+3*x*y(x)^2-y(x)^3),y(x), singsol=all)
\[ \frac {\ln \left (y \relax (x )\right )}{2}-\left (\int _{}^{\frac {x}{\sqrt {y \relax (x )}}-\sqrt {y \relax (x )}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}-\textit {\_a} +2}d \textit {\_a} \right )-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.648 (sec). Leaf size: 882
DSolve[y'[x] == y[x]^2/(x^3 + x^2*Sqrt[y[x]] - 3*x^2*y[x] + y[x]^(3/2) - 2*x*y[x]^(3/2) + y[x]^2 + 3*x*y[x]^2 + y[x]^(5/2) - y[x]^3),y[x],x,IncludeSingularSolutions -> True]
\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {-x-K[1]}{2 \left (-2 x^3+6 K[1] x^2-2 \sqrt {K[1]} x^2-6 K[1]^2 x+4 K[1]^{3/2} x+K[1] x+2 K[1]^3-2 K[1]^{5/2}-K[1]^2-2 K[1]^{3/2}\right )}+\text {RootSum}\left [2 K[1]^3-2 K[1]^{5/2}-6 \text {$\#$1} K[1]^2-K[1]^2+4 \text {$\#$1} K[1]^{3/2}-2 K[1]^{3/2}+6 \text {$\#$1}^2 K[1]+\text {$\#$1} K[1]-2 \text {$\#$1}^2 \sqrt {K[1]}-2 \text {$\#$1}^3\&,\frac {\log (x-\text {$\#$1})}{6 K[1]^2-4 K[1]^{3/2}-12 \text {$\#$1} K[1]-K[1]+4 \text {$\#$1} \sqrt {K[1]}+6 \text {$\#$1}^2}\&\right ]+\frac {\text {RootSum}\left [2 K[1]^3-2 K[1]^{5/2}-6 \text {$\#$1} K[1]^2-K[1]^2+4 \text {$\#$1} K[1]^{3/2}-2 K[1]^{3/2}+6 \text {$\#$1}^2 K[1]+\text {$\#$1} K[1]-2 \text {$\#$1}^2 \sqrt {K[1]}-2 \text {$\#$1}^3\&,\frac {10 \log (x-\text {$\#$1}) K[1]^3-10 K[1]^3-10 \log (x-\text {$\#$1}) K[1]^{5/2}-14 K[1]^{5/2}-10 x \log (x-\text {$\#$1}) K[1]^2-5 \log (x-\text {$\#$1}) K[1]^2-20 \log (x-\text {$\#$1}) \text {$\#$1} K[1]^2+20 \text {$\#$1} K[1]^2+8 K[1]^2-38 x \log (x-\text {$\#$1}) K[1]^{3/2}-10 \log (x-\text {$\#$1}) K[1]^{3/2}+58 \log (x-\text {$\#$1}) \text {$\#$1} K[1]^{3/2}-10 \text {$\#$1} K[1]^{3/2}+5 K[1]^{3/2}+10 \log (x-\text {$\#$1}) \text {$\#$1}^2 K[1]-10 \text {$\#$1}^2 K[1]+11 x \log (x-\text {$\#$1}) K[1]+20 x \log (x-\text {$\#$1}) \text {$\#$1} K[1]-6 \log (x-\text {$\#$1}) \text {$\#$1} K[1]+3 \text {$\#$1} K[1]-48 \log (x-\text {$\#$1}) \text {$\#$1}^2 \sqrt {K[1]}+24 \text {$\#$1}^2 \sqrt {K[1]}+38 x \log (x-\text {$\#$1}) \text {$\#$1} \sqrt {K[1]}-10 x \log (x-\text {$\#$1}) \text {$\#$1}^2}{-50 K[1]^3+50 K[1]^{5/2}+50 x K[1]^2+100 \text {$\#$1} K[1]^2+25 K[1]^2+190 x K[1]^{3/2}-290 \text {$\#$1} K[1]^{3/2}+50 K[1]^{3/2}-50 \text {$\#$1}^2 K[1]+259 x K[1]-100 x \text {$\#$1} K[1]-284 \text {$\#$1} K[1]+240 \text {$\#$1}^2 \sqrt {K[1]}-190 x \text {$\#$1} \sqrt {K[1]}+50 x \text {$\#$1}^2}\&\right ]}{K[1]}+\frac {1}{2 K[1]}\right )dK[1]-\text {RootSum}\left [-2 \text {$\#$1}^3+6 y(x) \text {$\#$1}^2-2 \sqrt {y(x)} \text {$\#$1}^2-6 y(x)^2 \text {$\#$1}+4 y(x)^{3/2} \text {$\#$1}+y(x) \text {$\#$1}+2 y(x)^3-2 y(x)^{5/2}-y(x)^2-2 y(x)^{3/2}\&,\frac {\log (x-\text {$\#$1})}{6 \text {$\#$1}^2-12 y(x) \text {$\#$1}+4 \sqrt {y(x)} \text {$\#$1}+6 y(x)^2-4 y(x)^{3/2}-y(x)}\&\right ] y(x)=c_1,y(x)\right ] \]