47.5.7 problem 7
Internal
problem
ID
[7496]
Book
:
Ordinary
differential
equations
and
calculus
of
variations.
Makarets
and
Reshetnyak.
Wold
Scientific.
Singapore.
1995
Section
:
Chapter
2.
Linear
homogeneous
equations.
Section
2.3.4
problems.
page
104
Problem
number
:
7
Date
solved
:
Sunday, March 30, 2025 at 12:10:43 PM
CAS
classification
:
[[_2nd_order, _linear, _nonhomogeneous]]
\begin{align*} x^{2} \left (x +1\right ) y^{\prime \prime }+x \left (4 x +3\right ) y^{\prime }-y&=x +\frac {1}{x} \end{align*}
✓ Maple. Time used: 0.659 (sec). Leaf size: 640
ode:=x^2*(1+x)*diff(diff(y(x),x),x)+x*(4*x+3)*diff(y(x),x)-y(x) = x+1/x;
dsolve(ode,y(x), singsol=all);
\begin{align*} \text {Solution too large to show}\end{align*}
✓ Mathematica. Time used: 7.806 (sec). Leaf size: 636
ode=x^2*(x+1)*D[y[x],{x,2}]+x*(4*x+3)*D[y[x],x]-y[x]==x+1/x;
ic={};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to x^{-1-\sqrt {2}} \left (x^{2 \sqrt {2}} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-x\right ) \int _1^x\frac {7 \operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[2]\right ) K[2]^{-1-\sqrt {2}} \left (K[2]^2+1\right )}{(K[2]+1) \left (\left (4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (-\sqrt {2},3-\sqrt {2},2-2 \sqrt {2},-K[2]\right ) \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[2]\right ) K[2]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[2]\right ) \left (14 \sqrt {2} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[2]\right )+\left (-4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (\sqrt {2},3+\sqrt {2},2 \left (1+\sqrt {2}\right ),-K[2]\right ) K[2]\right )\right )}dK[2]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-x\right ) \int _1^x-\frac {7 \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right ) K[1]^{-1+\sqrt {2}} \left (K[1]^2+1\right )}{(K[1]+1) \left (\left (4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (-\sqrt {2},3-\sqrt {2},2-2 \sqrt {2},-K[1]\right ) \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right ) K[1]+\operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-K[1]\right ) \left (14 \sqrt {2} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-K[1]\right )+\left (-4+\sqrt {2}\right ) \operatorname {Hypergeometric2F1}\left (\sqrt {2},3+\sqrt {2},2 \left (1+\sqrt {2}\right ),-K[1]\right ) K[1]\right )\right )}dK[1]+c_2 x^{2 \sqrt {2}} \operatorname {Hypergeometric2F1}\left (-1+\sqrt {2},2+\sqrt {2},1+2 \sqrt {2},-x\right )+c_1 \operatorname {Hypergeometric2F1}\left (-1-\sqrt {2},2-\sqrt {2},1-2 \sqrt {2},-x\right )\right )
\]
✗ Sympy
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*(x + 1)*Derivative(y(x), (x, 2)) + x*(4*x + 3)*Derivative(y(x), x) - x - y(x) - 1/x,0)
ics = {}
dsolve(ode,func=y(x),ics=ics)
NotImplementedError : The given ODE Derivative(y(x), x) - (-x**4*Derivative(y(x), (x, 2)) - x**3*Derivative(y(x), (x, 2)) + x**2 + x*y(x) + 1)/(x**2*(4*x + 3)) cannot be solved by the factorable group method