29.34.27 problem 1029
Internal
problem
ID
[5598]
Book
:
Ordinary
differential
equations
and
their
solutions.
By
George
Moseley
Murphy.
1960
Section
:
Various
34
Problem
number
:
1029
Date
solved
:
Sunday, March 30, 2025 at 09:05:39 AM
CAS
classification
:
[_quadrature]
\begin{align*} {y^{\prime }}^{3}-a x y^{\prime }+x^{3}&=0 \end{align*}
✓ Maple. Time used: 0.034 (sec). Leaf size: 248
ode:=diff(y(x),x)^3-a*x*diff(y(x),x)+x^3 = 0;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= -\frac {\int \left (\left (1+i \sqrt {3}\right ) \left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{1}/{3}}-\frac {12 x \left (i \sqrt {3}-1\right ) a}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{1}/{3}}}\right )d x}{12}+c_1 \\
y &= \frac {\int \left (\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )-\frac {12 \left (1+i \sqrt {3}\right ) x a}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{1}/{3}}}\right )d x}{12}+c_1 \\
y &= \frac {\int \frac {\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{2}/{3}}+12 a x}{\left (-108 x^{3}+12 \sqrt {3}\, \sqrt {-4 a^{3} x^{3}+27 x^{6}}\right )^{{1}/{3}}}d x}{6}+c_1 \\
\end{align*}
✓ Mathematica. Time used: 176.906 (sec). Leaf size: 349
ode=(D[y[x],x])^3 -a*x*D[y[x],x]+x^3==0;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*}
y(x)\to \int _1^x\frac {2 \sqrt [3]{3} a K[1]+\sqrt [3]{2} \left (\sqrt {81 K[1]^6-12 a^3 K[1]^3}-9 K[1]^3\right )^{2/3}}{6^{2/3} \sqrt [3]{\sqrt {81 K[1]^6-12 a^3 K[1]^3}-9 K[1]^3}}dK[1]+c_1 \\
y(x)\to \int _1^x\frac {i \sqrt [3]{3} \left (i+\sqrt {3}\right ) \left (2 \sqrt {81 K[2]^6-12 a^3 K[2]^3}-18 K[2]^3\right )^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (3 i+\sqrt {3}\right ) a K[2]}{12 \sqrt [3]{\sqrt {81 K[2]^6-12 a^3 K[2]^3}-9 K[2]^3}}dK[2]+c_1 \\
y(x)\to \int _1^x\frac {\sqrt [3]{3} \left (-1-i \sqrt {3}\right ) \left (2 \sqrt {81 K[3]^6-12 a^3 K[3]^3}-18 K[3]^3\right )^{2/3}-2 \sqrt [3]{2} \sqrt [6]{3} \left (-3 i+\sqrt {3}\right ) a K[3]}{12 \sqrt [3]{\sqrt {81 K[3]^6-12 a^3 K[3]^3}-9 K[3]^3}}dK[3]+c_1 \\
\end{align*}
✓ Sympy. Time used: 4.307 (sec). Leaf size: 335
from sympy import *
x = symbols("x")
a = symbols("a")
y = Function("y")
ode = Eq(-a*x*Derivative(y(x), x) + x**3 + Derivative(y(x), x)**3,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} - \frac {\sqrt [3]{18} a x^{2}}{6 \sqrt [3]{\sqrt {3} \sqrt {27 - 4 a^{3}} + 9}} - \frac {\sqrt [3]{12} x^{2} \sqrt [3]{\sqrt {3} \sqrt {27 - 4 a^{3}} + 9}}{12}, \ y{\left (x \right )} = C_{1} - \frac {\sqrt [3]{2} \cdot 3^{\frac {2}{3}} i a x^{2}}{3 \left (\sqrt {3} - i\right ) \sqrt [3]{\sqrt {3} \sqrt {27 - 4 a^{3}} + 9}} + \frac {2^{\frac {2}{3}} \cdot 3^{\frac {5}{6}} x^{2} \sqrt [3]{\sqrt {3} \sqrt {27 - 4 a^{3}} + 9}}{12 \left (\sqrt {3} - i\right )} + \frac {\sqrt [3]{12} i x^{2} \sqrt [3]{\sqrt {3} \sqrt {27 - 4 a^{3}} + 9}}{12 \left (\sqrt {3} - i\right )}, \ y{\left (x \right )} = C_{1} + \frac {\sqrt [3]{2} \cdot 3^{\frac {2}{3}} i a x^{2}}{3 \left (\sqrt {3} + i\right ) \sqrt [3]{\sqrt {3} \sqrt {27 - 4 a^{3}} + 9}} - \frac {\sqrt [3]{12} i x^{2} \sqrt [3]{\sqrt {3} \sqrt {27 - 4 a^{3}} + 9}}{12 \left (\sqrt {3} + i\right )} + \frac {2^{\frac {2}{3}} \cdot 3^{\frac {5}{6}} x^{2} \sqrt [3]{\sqrt {3} \sqrt {27 - 4 a^{3}} + 9}}{12 \left (\sqrt {3} + i\right )}\right ]
\]