78.10.2 problem 2
Internal
problem
ID
[18184]
Book
:
DIFFERENTIAL
EQUATIONS
WITH
APPLICATIONS
AND
HISTORICAL
NOTES
by
George
F.
Simmons.
3rd
edition.
2017.
CRC
press,
Boca
Raton
FL.
Section
:
Chapter
3.
Second
order
linear
equations.
Section
15.
The
General
Solution
of
the
Homogeneous
Equation.
Problems
at
page
117
Problem
number
:
2
Date
solved
:
Thursday, March 13, 2025 at 11:48:33 AM
CAS
classification
:
[[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\begin{align*} x^{2} y^{\prime \prime }-2 x y^{\prime }+2 y&=0 \end{align*}
With initial conditions
\begin{align*} y \left (1\right )&=3\\ y^{\prime }\left (1\right )&=5 \end{align*}
✓ Maple. Time used: 0.007 (sec). Leaf size: 11
ode:=x^2*diff(diff(y(x),x),x)-2*x*diff(y(x),x)+2*y(x) = 0;
ic:=y(1) = 3, D(y)(1) = 5;
dsolve([ode,ic],y(x), singsol=all);
\[
y = 2 x^{2}+x
\]
✓ Mathematica. Time used: 0.16 (sec). Leaf size: 438
ode=x^2*D[y[x],{x,2}] -2*D[y[x],x]+2*y[x]==0;
ic={y[1]==3,Derivative[1][y][1] == 5};
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
\[
y(x)\to \frac {x^{\frac {1}{2}-\frac {i \sqrt {7}}{2}} \left (\left (2 \left (3 \sqrt {7}+7 i\right ) \operatorname {Hypergeometric1F1}\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2},1-i \sqrt {7},-2\right )-3 \left (\sqrt {7}+3 i\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{2}-\frac {i \sqrt {7}}{2},2-i \sqrt {7},-2\right )\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{2} i \left (i+\sqrt {7}\right ),1+i \sqrt {7},-\frac {2}{x}\right )-\left (3 \left (\sqrt {7}-3 i\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{2}+\frac {i \sqrt {7}}{2},2+i \sqrt {7},-2\right )+2 \left (-3 \sqrt {7}+7 i\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{2} i \left (i+\sqrt {7}\right ),1+i \sqrt {7},-2\right )\right ) x^{i \sqrt {7}} \operatorname {Hypergeometric1F1}\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2},1-i \sqrt {7},-\frac {2}{x}\right )\right )}{\operatorname {Hypergeometric1F1}\left (-\frac {1}{2}-\frac {i \sqrt {7}}{2},1-i \sqrt {7},-2\right ) \left (4 \sqrt {7} \operatorname {Hypergeometric1F1}\left (\frac {1}{2} i \left (i+\sqrt {7}\right ),1+i \sqrt {7},-2\right )-\left (\left (\sqrt {7}-3 i\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{2}+\frac {i \sqrt {7}}{2},2+i \sqrt {7},-2\right )\right )\right )-\left (\left (\sqrt {7}+3 i\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{2}-\frac {i \sqrt {7}}{2},2-i \sqrt {7},-2\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{2} i \left (i+\sqrt {7}\right ),1+i \sqrt {7},-2\right )\right )}
\]
✓ Sympy. Time used: 0.155 (sec). Leaf size: 8
from sympy import *
x = symbols("x")
y = Function("y")
ode = Eq(x**2*Derivative(y(x), (x, 2)) - 2*x*Derivative(y(x), x) + 2*y(x),0)
ics = {y(1): 3, Subs(Derivative(y(x), x), x, 1): 5}
dsolve(ode,func=y(x),ics=ics)
\[
y{\left (x \right )} = x \left (2 x + 1\right )
\]