68.1.12 problem Problem 1.9
Internal
problem
ID
[14141]
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
:
Tuesday, January 28, 2025 at 06:15:49 AM
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*}
✓ Solution by Maple
Time used: 0.045 (sec). Leaf size: 158
Order:=6;
dsolve(2*x*diff(y(x),x$2)+(1+x)*diff(y(x),x)-k*y(x)=0,y(x),type='series',x=0);
\[
y = \sqrt {x}\, c_{1} \left (1+\left (\frac {k}{3}-\frac {1}{6}\right ) x +\left (\frac {1}{30} k^{2}-\frac {1}{15} k +\frac {1}{40}\right ) x^{2}+\frac {1}{5040} \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{3}+\frac {1}{362880} \left (2 k -7\right ) \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{4}+\frac {1}{39916800} \left (2 k -9\right ) \left (2 k -7\right ) \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+k x +\frac {1}{6} \left (k -1\right ) k x^{2}+\frac {1}{90} \left (-2+k \right ) \left (k -1\right ) k x^{3}+\frac {1}{2520} \left (k -3\right ) \left (-2+k \right ) \left (k -1\right ) k x^{4}+\frac {1}{113400} \left (k -4\right ) \left (k -3\right ) \left (-2+k \right ) \left (k -1\right ) k x^{5}+\operatorname {O}\left (x^{6}\right )\right )
\]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 304
AsymptoticDSolveValue[2*x*D[y[x],{x,2}]+(1+x)*D[y[x],x]-k*y[x]==0,y[x],{x,0,"6"-1}]
\[
y(x)\to c_1 \sqrt {x} \left (\frac {4 \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {5}{4}-\frac {k}{2}\right ) \left (\frac {7}{4}-\frac {k}{2}\right ) \left (\frac {9}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^5}{155925}-\frac {2 \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {5}{4}-\frac {k}{2}\right ) \left (\frac {7}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^4}{2835}+\frac {4}{315} \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {5}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^3-\frac {2}{15} \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^2+\frac {2}{3} \left (\frac {k}{2}-\frac {1}{4}\right ) x+1\right )+c_2 \left (\frac {2 \left (\frac {1}{2}-\frac {k}{2}\right ) \left (1-\frac {k}{2}\right ) \left (\frac {3}{2}-\frac {k}{2}\right ) \left (2-\frac {k}{2}\right ) k x^5}{14175}-\frac {1}{315} \left (\frac {1}{2}-\frac {k}{2}\right ) \left (1-\frac {k}{2}\right ) \left (\frac {3}{2}-\frac {k}{2}\right ) k x^4+\frac {2}{45} \left (\frac {1}{2}-\frac {k}{2}\right ) \left (1-\frac {k}{2}\right ) k x^3-\frac {1}{3} \left (\frac {1}{2}-\frac {k}{2}\right ) k x^2+k x+1\right )
\]