Internal problem ID [8335]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 1, Additional non-linear first order
Problem number: 755.
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}}}{y^{\frac {3}{2}}+x^{2}-2 x y+y^{2}}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.043 (sec). Leaf size: 84
dsolve(diff(y(x),x) = y(x)^(3/2)/(y(x)^(3/2)+x^2-2*x*y(x)+y(x)^2),y(x), singsol=all)
\[ \frac {4 y \relax (x )}{\left (-x +y \relax (x )\right )^{2}}+\frac {4 \sqrt {y \relax (x )}}{\left (-x +y \relax (x )\right )^{2}}-\frac {8 x}{\left (-x +y \relax (x )\right )^{2}}+\frac {1}{\left (-x +y \relax (x )\right )^{2}}-\frac {4 x}{\left (-x +y \relax (x )\right )^{2} \sqrt {y \relax (x )}}+\frac {4 x^{2}}{\left (-x +y \relax (x )\right )^{2} y \relax (x )}-c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 156.068 (sec). Leaf size: 2358
DSolve[y'[x] == y[x]^(3/2)/(x^2 - 2*x*y[x] + y[x]^(3/2) + y[x]^2),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {1}{3} \left (-\sqrt [3]{x^3+3 e^{c_1} x^2+6 \sqrt {3} \sqrt {-e^{4 c_1} \left (x^3+3 e^{c_1} x^2-e^{2 c_1} (8 x-3) x+e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)+e^{3 c_1} (12 x+1)-96 e^{5 c_1}-64 e^{6 c_1}}+\frac {-x^2-2 e^{c_1} x+e^{2 c_1} (8 x-1)-16 e^{3 c_1}-16 e^{4 c_1}}{\sqrt [3]{x^3+3 e^{c_1} x^2+6 \sqrt {3} \sqrt {-e^{4 c_1} \left (x^3+3 e^{c_1} x^2-e^{2 c_1} (8 x-3) x+e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)+e^{3 c_1} (12 x+1)-96 e^{5 c_1}-64 e^{6 c_1}}}+2 \left (x+e^{c_1}+2 e^{2 c_1}\right )\right ) \\ y(x)\to \frac {1}{6} \left (\left (1-i \sqrt {3}\right ) \sqrt [3]{x^3+3 e^{c_1} x^2+6 \sqrt {3} \sqrt {-e^{4 c_1} \left (x^3+3 e^{c_1} x^2-e^{2 c_1} (8 x-3) x+e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)+e^{3 c_1} (12 x+1)-96 e^{5 c_1}-64 e^{6 c_1}}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2+2 e^{c_1} x+e^{2 c_1} (1-8 x)+16 e^{3 c_1}+16 e^{4 c_1}\right )}{\sqrt [3]{x^3+3 e^{c_1} x^2+6 \sqrt {3} \sqrt {-e^{4 c_1} \left (x^3+3 e^{c_1} x^2-e^{2 c_1} (8 x-3) x+e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)+e^{3 c_1} (12 x+1)-96 e^{5 c_1}-64 e^{6 c_1}}}+4 \left (x+e^{c_1}+2 e^{2 c_1}\right )\right ) \\ y(x)\to \frac {1}{6} \left (\left (1+i \sqrt {3}\right ) \sqrt [3]{x^3+3 e^{c_1} x^2+6 \sqrt {3} \sqrt {-e^{4 c_1} \left (x^3+3 e^{c_1} x^2-e^{2 c_1} (8 x-3) x+e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)+e^{3 c_1} (12 x+1)-96 e^{5 c_1}-64 e^{6 c_1}}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2+2 e^{c_1} x+e^{2 c_1} (1-8 x)+16 e^{3 c_1}+16 e^{4 c_1}\right )}{\sqrt [3]{x^3+3 e^{c_1} x^2+6 \sqrt {3} \sqrt {-e^{4 c_1} \left (x^3+3 e^{c_1} x^2-e^{2 c_1} (8 x-3) x+e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)+e^{3 c_1} (12 x+1)-96 e^{5 c_1}-64 e^{6 c_1}}}+4 \left (x+e^{c_1}+2 e^{2 c_1}\right )\right ) \\ y(x)\to \frac {1}{3} \left (-\sqrt [3]{x^3-3 e^{c_1} x^2+6 \sqrt {3} \sqrt {e^{4 c_1} \left (-x^3+3 e^{c_1} x^2+e^{2 c_1} (8 x-3) x-e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)-e^{3 c_1} (12 x+1)+96 e^{5 c_1}-64 e^{6 c_1}}+\frac {-x^2+2 e^{c_1} x+e^{2 c_1} (8 x-1)+16 e^{3 c_1}-16 e^{4 c_1}}{\sqrt [3]{x^3-3 e^{c_1} x^2+6 \sqrt {3} \sqrt {e^{4 c_1} \left (-x^3+3 e^{c_1} x^2+e^{2 c_1} (8 x-3) x-e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)-e^{3 c_1} (12 x+1)+96 e^{5 c_1}-64 e^{6 c_1}}}+2 \left (x-e^{c_1}+2 e^{2 c_1}\right )\right ) \\ y(x)\to \frac {1}{6} \left (\left (1-i \sqrt {3}\right ) \sqrt [3]{x^3-3 e^{c_1} x^2+6 \sqrt {3} \sqrt {e^{4 c_1} \left (-x^3+3 e^{c_1} x^2+e^{2 c_1} (8 x-3) x-e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)-e^{3 c_1} (12 x+1)+96 e^{5 c_1}-64 e^{6 c_1}}+\frac {\left (1+i \sqrt {3}\right ) \left (x^2-2 e^{c_1} x+e^{2 c_1} (1-8 x)-16 e^{3 c_1}+16 e^{4 c_1}\right )}{\sqrt [3]{x^3-3 e^{c_1} x^2+6 \sqrt {3} \sqrt {e^{4 c_1} \left (-x^3+3 e^{c_1} x^2+e^{2 c_1} (8 x-3) x-e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)-e^{3 c_1} (12 x+1)+96 e^{5 c_1}-64 e^{6 c_1}}}+4 \left (x-e^{c_1}+2 e^{2 c_1}\right )\right ) \\ y(x)\to \frac {1}{6} \left (\left (1+i \sqrt {3}\right ) \sqrt [3]{x^3-3 e^{c_1} x^2+6 \sqrt {3} \sqrt {e^{4 c_1} \left (-x^3+3 e^{c_1} x^2+e^{2 c_1} (8 x-3) x-e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)-e^{3 c_1} (12 x+1)+96 e^{5 c_1}-64 e^{6 c_1}}+\frac {\left (1-i \sqrt {3}\right ) \left (x^2-2 e^{c_1} x+e^{2 c_1} (1-8 x)-16 e^{3 c_1}+16 e^{4 c_1}\right )}{\sqrt [3]{x^3-3 e^{c_1} x^2+6 \sqrt {3} \sqrt {e^{4 c_1} \left (-x^3+3 e^{c_1} x^2+e^{2 c_1} (8 x-3) x-e^{4 c_1} (16 x+1)+e^{3 c_1} (20 x+1)\right )}-3 e^{2 c_1} (4 x-1) x+6 e^{4 c_1} (8 x-5)-e^{3 c_1} (12 x+1)+96 e^{5 c_1}-64 e^{6 c_1}}}+4 \left (x-e^{c_1}+2 e^{2 c_1}\right )\right ) \\ y(x)\to 0 \\ y(x)\to -\frac {\left (x-\sqrt [3]{x^3}\right )^2}{3 \sqrt [3]{x^3}} \\ y(x)\to \frac {\left (1+i \sqrt {3}\right ) \sqrt [3]{x^3} x+\left (1-i \sqrt {3}\right ) \left (x^3\right )^{2/3}+4 x^2}{6 x} \\ y(x)\to \frac {\left (1-i \sqrt {3}\right ) \sqrt [3]{x^3} x+\left (1+i \sqrt {3}\right ) \left (x^3\right )^{2/3}+4 x^2}{6 x} \\ \end{align*}