50.3.19 problem 19
Internal
problem
ID
[10163]
Book
:
Own
collection
of
miscellaneous
problems
Section
:
section
3.0
Problem
number
:
19
Date
solved
:
Tuesday, September 30, 2025 at 07:05:29 PM
CAS
classification
:
[[_3rd_order, _with_linear_symmetries]]
\begin{align*} x^{4} y^{\prime \prime \prime }+x^{3} y^{\prime \prime }+x^{2} y^{\prime }+x y&=x \end{align*}
✓ Maple. Time used: 0.006 (sec). Leaf size: 223
ode:=x^4*diff(diff(diff(y(x),x),x),x)+x^3*diff(diff(y(x),x),x)+x^2*diff(y(x),x)+x*y(x) = x;
dsolve(ode,y(x), singsol=all);
\[
y = c_2 \,x^{\frac {\left (-3 \sqrt {249}+47\right ) \left (188+12 \sqrt {249}\right )^{{2}/{3}}}{192}+\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}+\frac {2}{3}} \cos \left (\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \left (3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {83}-47 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+16\right ) \ln \left (x \right )}{192}\right )+c_3 \,x^{\frac {\left (-3 \sqrt {249}+47\right ) \left (188+12 \sqrt {249}\right )^{{2}/{3}}}{192}+\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{12}+\frac {2}{3}} \sin \left (\frac {\left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \left (3 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}} \sqrt {3}\, \sqrt {83}-47 \left (188+12 \sqrt {3}\, \sqrt {83}\right )^{{1}/{3}}+16\right ) \ln \left (x \right )}{192}\right )+c_1 \,x^{\frac {\left (3 \sqrt {249}-47\right ) \left (188+12 \sqrt {249}\right )^{{2}/{3}}}{96}-\frac {\left (188+12 \sqrt {249}\right )^{{1}/{3}}}{6}+\frac {2}{3}}+1
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 82
ode=x^4*D[y[x],{x,3}]+x^3*D[y[x],{x,2}]+x^2*D[y[x],x]+x*y[x]== x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\begin{align*} y(x)&\to c_1 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,1\right ]}+c_3 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,3\right ]}+c_2 x^{\text {Root}\left [\text {$\#$1}^3-2 \text {$\#$1}^2+2 \text {$\#$1}+1\&,2\right ]}+1 \end{align*}
✓ Sympy. Time used: 0.903 (sec). Leaf size: 233
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**4*Derivative(y(x), (x, 3)) + x**3*Derivative(y(x), (x, 2)) + x**2*Derivative(y(x), x) + x*y(x) - x,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = \frac {C_{1}}{x^{- \frac {2}{3} - \frac {2 \sqrt [3]{2}}{3 \sqrt [3]{47 + 3 \sqrt {249}}} + \frac {2^{\frac {2}{3}} \sqrt [3]{47 + 3 \sqrt {249}}}{6}}} + C_{2} x^{- \frac {\sqrt [3]{2}}{3 \sqrt [3]{47 + 3 \sqrt {249}}} + \frac {2^{\frac {2}{3}} \sqrt [3]{47 + 3 \sqrt {249}}}{12} + \frac {2}{3}} \sin {\left (\frac {\sqrt [3]{2} \sqrt {3} \left (\frac {4}{\sqrt [3]{47 + 3 \sqrt {249}}} + \sqrt [3]{2} \sqrt [3]{47 + 3 \sqrt {249}}\right ) \log {\left (x \right )}}{12} \right )} + C_{3} x^{- \frac {\sqrt [3]{2}}{3 \sqrt [3]{47 + 3 \sqrt {249}}} + \frac {2^{\frac {2}{3}} \sqrt [3]{47 + 3 \sqrt {249}}}{12} + \frac {2}{3}} \cos {\left (\frac {\sqrt [3]{2} \sqrt {3} \left (\frac {4}{\sqrt [3]{47 + 3 \sqrt {249}}} + \sqrt [3]{2} \sqrt [3]{47 + 3 \sqrt {249}}\right ) \log {\left (x \right )}}{12} \right )} + 1
\]