60.1.305 problem 306

Internal problem ID [10319]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 306
Date solved : Monday, January 27, 2025 at 07:08:20 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} \left (y^{3}-x^{3}\right ) y^{\prime }-x^{2} y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.909 (sec). Leaf size: 381

dsolve((y(x)^3-x^3)*diff(y(x),x)-x^2*y(x) = 0,y(x), singsol=all)
 

\begin{align*} y &= \frac {x}{\left (x^{3} c_{1} \left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right )\right )^{{1}/{3}}} \\ y &= \frac {x}{{\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\ y &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} \left (x^{3} c_{1} \left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right )\right )^{{1}/{3}}} \\ y &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} \left (x^{3} c_{1} \left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right )\right )^{{1}/{3}}} \\ y &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} \left (x^{3} c_{1} \left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right )\right )^{{1}/{3}}} \\ y &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} \left (x^{3} c_{1} \left (-c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right )\right )^{{1}/{3}}} \\ y &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\ y &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\ y &= \frac {4 x}{\left (i \sqrt {3}-1\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\ y &= \frac {4 x}{\left (1+i \sqrt {3}\right )^{2} {\left (-\left (c_{1} x^{3}+\sqrt {c_{1}^{2} x^{6}+1}\right ) c_{1} x^{3}\right )}^{{1}/{3}}} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.113 (sec). Leaf size: 48

DSolve[-(x^2*y[x]) + (-x^3 + y[x]^3)*D[y[x],x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(x)}{x}}\frac {(K[1]-1) \left (K[1]^2+K[1]+1\right )}{K[1] \left (K[1]^3-2\right )}dK[1]=-\log (x)+c_1,y(x)\right ] \]