68.1.12 problem Problem 1.9
Internal
problem
ID
[14062]
Book
:
Differential
Equations,
Linear,
Nonlinear,
Ordinary,
Partial.
A.C.
King,
J.Billingham,
S.R.Otto.
Cambridge
Univ.
Press
2003
Section
:
Chapter
1
VARIABLE
COEFFICIENT,
SECOND
ORDER
DIFFERENTIAL
EQUATIONS.
Problems
page
28
Problem
number
:
Problem
1.9
Date
solved
:
Wednesday, March 05, 2025 at 10:28:00 PM
CAS
classification
:
[[_2nd_order, _with_linear_symmetries]]
\begin{align*} 2 x y^{\prime \prime }+\left (1+x \right ) y^{\prime }-k y&=0 \end{align*}
Using series method with expansion around
\begin{align*} 0 \end{align*}
✓ Maple. Time used: 0.045 (sec). Leaf size: 158
Order:=6;
ode:=2*x*diff(diff(y(x),x),x)+(1+x)*diff(y(x),x)-k*y(x) = 0;
dsolve(ode,y(x),type='series',x=0);
\[
y = \sqrt {x}\, c_{1} \left (1+\left (\frac {k}{3}-\frac {1}{6}\right ) x +\left (\frac {1}{30} k^{2}-\frac {1}{15} k +\frac {1}{40}\right ) x^{2}+\frac {1}{5040} \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{3}+\frac {1}{362880} \left (2 k -7\right ) \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{4}+\frac {1}{39916800} \left (2 k -9\right ) \left (2 k -7\right ) \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+k x +\frac {1}{6} \left (k -1\right ) k x^{2}+\frac {1}{90} \left (-2+k \right ) \left (k -1\right ) k x^{3}+\frac {1}{2520} \left (k -3\right ) \left (-2+k \right ) \left (k -1\right ) k x^{4}+\frac {1}{113400} \left (k -4\right ) \left (k -3\right ) \left (-2+k \right ) \left (k -1\right ) k x^{5}+\operatorname {O}\left (x^{6}\right )\right )
\]
✓ Mathematica. Time used: 0.003 (sec). Leaf size: 304
ode=2*x*D[y[x],{x,2}]+(1+x)*D[y[x],x]-k*y[x]==0;
ic={};
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
\[
y(x)\to c_1 \sqrt {x} \left (\frac {4 \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {5}{4}-\frac {k}{2}\right ) \left (\frac {7}{4}-\frac {k}{2}\right ) \left (\frac {9}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^5}{155925}-\frac {2 \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {5}{4}-\frac {k}{2}\right ) \left (\frac {7}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^4}{2835}+\frac {4}{315} \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {5}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^3-\frac {2}{15} \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^2+\frac {2}{3} \left (\frac {k}{2}-\frac {1}{4}\right ) x+1\right )+c_2 \left (\frac {2 \left (\frac {1}{2}-\frac {k}{2}\right ) \left (1-\frac {k}{2}\right ) \left (\frac {3}{2}-\frac {k}{2}\right ) \left (2-\frac {k}{2}\right ) k x^5}{14175}-\frac {1}{315} \left (\frac {1}{2}-\frac {k}{2}\right ) \left (1-\frac {k}{2}\right ) \left (\frac {3}{2}-\frac {k}{2}\right ) k x^4+\frac {2}{45} \left (\frac {1}{2}-\frac {k}{2}\right ) \left (1-\frac {k}{2}\right ) k x^3-\frac {1}{3} \left (\frac {1}{2}-\frac {k}{2}\right ) k x^2+k x+1\right )
\]
✓ Sympy. Time used: 1.830 (sec). Leaf size: 158
from sympy import *
x = symbols("x")
k = symbols("k")
y = Function("y")
ode = Eq(-k*y(x) + 2*x*Derivative(y(x), (x, 2)) + (x + 1)*Derivative(y(x), x),0)
ics = {}
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
\[
y{\left (x \right )} = C_{2} \left (\frac {k x^{5} \left (1 - k\right ) \left (1 - \frac {k}{2}\right ) \left (2 - \frac {k}{2}\right ) \left (3 - k\right )}{28350} - \frac {k x^{4} \left (1 - k\right ) \left (1 - \frac {k}{2}\right ) \left (3 - k\right )}{1260} + \frac {k x^{3} \left (1 - k\right ) \left (1 - \frac {k}{2}\right )}{45} - \frac {k x^{2} \left (1 - k\right )}{6} + k x + 1\right ) + C_{1} \sqrt {x} \left (\frac {x^{4} \left (1 - 2 k\right ) \left (3 - 2 k\right ) \left (5 - 2 k\right ) \left (7 - 2 k\right )}{362880} - \frac {x^{3} \left (1 - 2 k\right ) \left (3 - 2 k\right ) \left (5 - 2 k\right )}{5040} + \frac {x^{2} \left (1 - 2 k\right ) \left (3 - 2 k\right )}{120} - \frac {x \left (1 - 2 k\right )}{6} + 1\right ) + O\left (x^{6}\right )
\]