66.2.35 problem Problem 50
Internal
problem
ID
[13855]
Book
:
Differential
equations
and
the
calculus
of
variations
by
L.
ElSGOLTS.
MIR
PUBLISHERS,
MOSCOW,
Third
printing
1977.
Section
:
Chapter
2,
DIFFERENTIAL
EQUATIONS
OF
THE
SECOND
ORDER
AND
HIGHER.
Problems
page
172
Problem
number
:
Problem
50
Date
solved
:
Wednesday, March 05, 2025 at 10:19:45 PM
CAS
classification
:
[[_2nd_order, _quadrature]]
\begin{align*} {y^{\prime \prime }}^{3}+y^{\prime \prime }+1&=x \end{align*}
✓ Maple. Time used: 0.069 (sec). Leaf size: 222
ode:=diff(diff(y(x),x),x)^3+diff(diff(y(x),x),x)+1 = x;
dsolve(ode,y(x), singsol=all);
\begin{align*}
y &= \frac {\left (\int \left (\int \frac {\left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{{2}/{3}}-12}{\left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{{1}/{3}}}d x \right )d x \right )}{6}+c_{1} x +c_{2} \\
y &= -\frac {\left (\int \left (\int \frac {i \sqrt {3}\, \left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{{2}/{3}}+12 i \sqrt {3}+\left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{{2}/{3}}-12}{\left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{{1}/{3}}}d x \right )d x \right )}{12}+c_{1} x +c_{2} \\
y &= \frac {\left (\int \left (\int \frac {\left (i \sqrt {3}-1\right ) \left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{{2}/{3}}+12 i \sqrt {3}+12}{\left (-108+108 x +12 \sqrt {81 x^{2}-162 x +93}\right )^{{1}/{3}}}d x \right )d x \right )}{12}+c_{1} x +c_{2} \\
\end{align*}
✗ Mathematica
ode=D[y[x],{x,2}]^3+D[y[x],{x,2}]+1==x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
Timed out
✓ Sympy. Time used: 56.347 (sec). Leaf size: 371
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(-x + Derivative(y(x), (x, 2))**3 + Derivative(y(x), (x, 2)) + 1,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
\left [ y{\left (x \right )} = C_{1} + \frac {6 \sqrt {3} C_{2} x - 6 i C_{2} x + 4 \sqrt [3]{18} i \iint \frac {1}{\sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 54 x + 31} + 9}}\, dx\, dx + 2^{\frac {2}{3}} \cdot 3^{\frac {5}{6}} \iint \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 54 x + 31} + 9}\, dx\, dx + \sqrt [3]{12} i \iint \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 54 x + 31} + 9}\, dx\, dx}{6 \left (\sqrt {3} - i\right )}, \ y{\left (x \right )} = C_{1} + \frac {6 \sqrt {3} C_{2} x + 6 i C_{2} x - 4 \sqrt [3]{18} i \iint \frac {1}{\sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 54 x + 31} + 9}}\, dx\, dx + 2^{\frac {2}{3}} \cdot 3^{\frac {5}{6}} \iint \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 54 x + 31} + 9}\, dx\, dx - \sqrt [3]{12} i \iint \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 54 x + 31} + 9}\, dx\, dx}{6 \left (\sqrt {3} + i\right )}, \ y{\left (x \right )} = C_{1} + C_{2} x + \frac {\int \left (2 \sqrt [3]{18} \int \frac {1}{\sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 54 x + 31} + 9}}\, dx - \sqrt [3]{12} \int \sqrt [3]{- 9 x + \sqrt {3} \sqrt {27 x^{2} - 54 x + 31} + 9}\, dx\right )\, dx}{6}\right ]
\]