60.2.269 problem 845

Internal problem ID [10856]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, Additional non-linear first order
Problem number : 845
Date solved : Tuesday, January 28, 2025 at 05:24:31 PM
CAS classification : [NONE]

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

✓ Solution by Maple

Time used: 0.020 (sec). Leaf size: 44

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

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

✓ Solution by Mathematica

Time used: 4.547 (sec). Leaf size: 218

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

\begin{align*} y(x)\to -\frac {1}{2} \sqrt [3]{-\frac {1}{2}} \sqrt [3]{9 x^8+24 x^7+16 x^6+72 x^5+12 (11+6 c_1) x^4+96 c_1 x^3+144 x^2+288 c_1 x+144 c_1{}^2} \\ y(x)\to \frac {1}{2} \sqrt [3]{\frac {9 x^8}{2}+12 x^7+8 x^6+36 x^5+6 (11+6 c_1) x^4+48 c_1 x^3+72 x^2+144 c_1 x+72 c_1{}^2} \\ y(x)\to \frac {1}{2} (-1)^{2/3} \sqrt [3]{\frac {9 x^8}{2}+12 x^7+8 x^6+36 x^5+6 (11+6 c_1) x^4+48 c_1 x^3+72 x^2+144 c_1 x+72 c_1{}^2} \\ \end{align*}