Internal problem ID [8499]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 919.
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^{\frac {3}{2}} \left (x -y+\sqrt {y}\right )}{y^{\frac {3}{2}} x -y^{\frac {5}{2}}+y^{2}+x^{3}-3 x^{2} y+3 x y^{2}-y^{3}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 397
dsolve(diff(y(x),x) = y(x)^(3/2)*(x-y(x)+y(x)^(1/2))/(y(x)^(3/2)*x-y(x)^(5/2)+y(x)^2+x^3-3*x^2*y(x)+3*x*y(x)^2-y(x)^3),y(x), singsol=all)
\[ -c_{1}+\frac {1}{\left (-x +y \relax (x )\right )^{6}}-\frac {6 y \relax (x )}{\left (-x +y \relax (x )\right )^{6}}+\frac {9 y \relax (x )^{2}}{\left (-x +y \relax (x )\right )^{6}}+\frac {4 y \relax (x )^{3}}{\left (-x +y \relax (x )\right )^{6}}-\frac {4 y \relax (x )^{\frac {3}{2}}}{\left (-x +y \relax (x )\right )^{6}}+\frac {12 y \relax (x )^{\frac {5}{2}}}{\left (-x +y \relax (x )\right )^{6}}-\frac {6 x^{2}}{\left (-x +y \relax (x )\right )^{6} y \relax (x )}+\frac {60 x^{4}}{\left (-x +y \relax (x )\right )^{6} y \relax (x )}+\frac {12 x}{\left (-x +y \relax (x )\right )^{6}}-\frac {80 x^{3}}{\left (-x +y \relax (x )\right )^{6}}-\frac {36 x^{3}}{\left (-x +y \relax (x )\right )^{6} y \relax (x )}+\frac {4 x^{6}}{\left (-x +y \relax (x )\right )^{6} y \relax (x )^{3}}+\frac {60 y \relax (x ) x^{2}}{\left (-x +y \relax (x )\right )^{6}}-\frac {24 y \relax (x )^{2} x}{\left (-x +y \relax (x )\right )^{6}}-\frac {24 x^{5}}{\left (-x +y \relax (x )\right )^{6} y \relax (x )^{2}}+\frac {54 x^{2}}{\left (-x +y \relax (x )\right )^{6}}+\frac {9 x^{4}}{\left (-x +y \relax (x )\right )^{6} y \relax (x )^{2}}-\frac {36 y \relax (x ) x}{\left (-x +y \relax (x )\right )^{6}}+\frac {60 x^{4}}{\left (-x +y \relax (x )\right )^{6} y \relax (x )^{\frac {3}{2}}}-\frac {12 x^{5}}{\left (-x +y \relax (x )\right )^{6} y \relax (x )^{\frac {5}{2}}}+\frac {12 \sqrt {y \relax (x )}\, x}{\left (-x +y \relax (x )\right )^{6}}+\frac {120 \sqrt {y \relax (x )}\, x^{2}}{\left (-x +y \relax (x )\right )^{6}}-\frac {120 x^{3}}{\left (-x +y \relax (x )\right )^{6} \sqrt {y \relax (x )}}-\frac {60 y \relax (x )^{\frac {3}{2}} x}{\left (-x +y \relax (x )\right )^{6}}-\frac {12 x^{2}}{\left (-x +y \relax (x )\right )^{6} \sqrt {y \relax (x )}}+\frac {4 x^{3}}{\left (-x +y \relax (x )\right )^{6} y \relax (x )^{\frac {3}{2}}} = 0 \]
✓ Solution by Mathematica
Time used: 53.27 (sec). Leaf size: 251
DSolve[y'[x] == ((x + Sqrt[y[x]] - y[x])*y[x]^(3/2))/(x^3 - 3*x^2*y[x] + 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]
\begin{align*} y(x)\to \text {Root}\left [\text {$\#$1}^9 c_1{}^4-6 \text {$\#$1}^8 c_1{}^4 x+\text {$\#$1}^7 \left (15 c_1{}^4 x^2-6 c_1{}^2\right )+\text {$\#$1}^6 \left (-20 c_1{}^4 x^3+30 c_1{}^2 x-4+2 c_1{}^2\right )+\text {$\#$1}^5 \left (15 c_1{}^4 x^4-60 c_1{}^2 x^2+24 x-6 c_1{}^2 x+9\right )+\text {$\#$1}^4 \left (-6 c_1{}^4 x^5+60 c_1{}^2 x^3-60 x^2+6 c_1{}^2 x^2-36 x-6\right )+\text {$\#$1}^3 \left (c_1{}^4 x^6-30 c_1{}^2 x^4+80 x^3-2 c_1{}^2 x^3+54 x^2+12 x+1\right )+\text {$\#$1}^2 \left (6 c_1{}^2 x^5-60 x^4-36 x^3-6 x^2\right )+\text {$\#$1} \left (24 x^5+9 x^4\right )-4 x^6\&,1\right ] \\ \end{align*}