60.2.144 problem 720

Internal problem ID [10731]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 720
Date solved : Tuesday, January 28, 2025 at 05:09:31 PM
CAS classification : [`y=_G(x,y')`]

\begin{align*} y^{\prime }&=\frac {x^{3} \left (3 x +3+\sqrt {9 x^{4}-4 y^{3}}\right )}{\left (1+x \right ) y^{2}} \end{align*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 48

dsolve(diff(y(x),x) = x^3*(3*x+3+(9*x^4-4*y(x)^3)^(1/2))/(x+1)/y(x)^2,y(x), singsol=all)
 

\[ \int _{\textit {\_b}}^{y}\frac {\textit {\_a}^{2}}{\sqrt {9 x^{4}-4 \textit {\_a}^{3}}}d \textit {\_a} -\frac {x^{3}}{3}+\frac {x^{2}}{2}-x +\ln \left (x +1\right )-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 4.852 (sec). Leaf size: 175

DSolve[D[y[x],x] == (x^3*(3 + 3*x + Sqrt[9*x^4 - 4*y[x]^3]))/((1 + x)*y[x]^2),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to \left (-\frac {3}{2}\right )^{2/3} \sqrt [3]{x^4-4 \left (\int \frac {x^3}{x+1} \, dx\right )^2+8 c_1 \int \frac {x^3}{x+1} \, dx-4 c_1{}^2} \\ y(x)\to \left (\frac {3}{2}\right )^{2/3} \sqrt [3]{x^4-4 \left (\int \frac {x^3}{x+1} \, dx\right )^2+8 c_1 \int \frac {x^3}{x+1} \, dx-4 c_1{}^2} \\ y(x)\to -\sqrt [3]{-1} \left (\frac {3}{2}\right )^{2/3} \sqrt [3]{x^4-4 \left (\int \frac {x^3}{x+1} \, dx\right )^2+8 c_1 \int \frac {x^3}{x+1} \, dx-4 c_1{}^2} \\ \end{align*}