60.2.179 problem 755

Internal problem ID [10766]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 755
Date solved : Monday, January 27, 2025 at 09:43:05 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

\begin{align*} y^{\prime }&=\frac {y^{{3}/{2}}}{y^{{3}/{2}}+x^{2}-2 y x +y^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.061 (sec). Leaf size: 71

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)
 

\[ \frac {4 \sqrt {y}\, x^{2}-y^{{7}/{2}} c_{1} +\left (2 c_{1} x +4\right ) y^{{5}/{2}}+4 y^{2}+\left (-c_{1} x^{2}-8 x +1\right ) y^{{3}/{2}}-4 x y}{\left (x -y\right )^{2} y^{{3}/{2}}} = 0 \]

✓ Solution by Mathematica

Time used: 60.232 (sec). Leaf size: 2213

DSolve[D[y[x],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 ) \\ \end{align*}