60.2.345 problem 922

Internal problem ID [10932]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 922
Date solved : Monday, January 27, 2025 at 10:25:29 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _rational]

\begin{align*} y^{\prime }&=\frac {y^{2}}{y^{2}+y^{{3}/{2}}+\sqrt {y}\, x^{2}-2 y^{{3}/{2}} x +y^{{5}/{2}}+x^{3}-3 x^{2} y+3 x y^{2}-y^{3}} \end{align*}

✓ Solution by Maple

Time used: 0.269 (sec). Leaf size: 45

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)
 

\[ \frac {\ln \left (y\right )}{2}-\int _{}^{\frac {x -y}{\sqrt {y}}}\frac {1}{2 \textit {\_a}^{3}+2 \textit {\_a}^{2}-\textit {\_a} +2}d \textit {\_a} -c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.940 (sec). Leaf size: 437

DSolve[D[y[x],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]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (\frac {-x-K[2]}{2 \left (-2 x^3+6 K[2] x^2-2 \sqrt {K[2]} x^2-6 K[2]^2 x+4 K[2]^{3/2} x+K[2] x+2 K[2]^3-2 K[2]^{5/2}-K[2]^2-2 K[2]^{3/2}\right )}-\int _1^x\left (\frac {K[2] \left (\frac {K[1]^2}{\sqrt {K[2]}}-6 K[1]^2+12 K[2] K[1]-6 \sqrt {K[2]} K[1]-K[1]-6 K[2]^2+5 K[2]^{3/2}+2 K[2]+3 \sqrt {K[2]}\right )}{\left (2 K[1]^3-6 K[2] K[1]^2+2 \sqrt {K[2]} K[1]^2+6 K[2]^2 K[1]-4 K[2]^{3/2} K[1]-K[2] K[1]-2 K[2]^3+2 K[2]^{5/2}+K[2]^2+2 K[2]^{3/2}\right )^2}-\frac {1}{2 K[1]^3-6 K[2] K[1]^2+2 \sqrt {K[2]} K[1]^2+6 K[2]^2 K[1]-4 K[2]^{3/2} K[1]-K[2] K[1]-2 K[2]^3+2 K[2]^{5/2}+K[2]^2+2 K[2]^{3/2}}\right )dK[1]+\frac {1}{2 K[2]}\right )dK[2]+\int _1^x-\frac {y(x)}{2 K[1]^3-6 y(x) K[1]^2+2 \sqrt {y(x)} K[1]^2+6 y(x)^2 K[1]-4 y(x)^{3/2} K[1]-y(x) K[1]-2 y(x)^3+2 y(x)^{5/2}+y(x)^2+2 y(x)^{3/2}}dK[1]=c_1,y(x)\right ] \]