Internal problem ID [12405]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {2 x y^{\prime \prime }+\left (x +1\right ) y^{\prime }-y k=0} \] With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 132
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);
\[ y \left (x \right ) = \sqrt {x}\, c_{1} \left (1+\left (\frac {k}{3}-\frac {1}{6}\right ) x +\left (\frac {1}{30} k^{2}-\frac {1}{15} k +\frac {1}{40}\right ) x^{2}+\frac {1}{5040} \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{3}+\frac {1}{362880} \left (2 k -7\right ) \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{4}+\frac {1}{39916800} \left (2 k -9\right ) \left (2 k -7\right ) \left (2 k -5\right ) \left (2 k -3\right ) \left (-1+2 k \right ) x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1+k x +\frac {1}{6} \left (-1+k \right ) k x^{2}+\frac {1}{90} \left (-2+k \right ) \left (-1+k \right ) k x^{3}+\frac {1}{2520} \left (k -3\right ) \left (-2+k \right ) \left (-1+k \right ) k x^{4}+\frac {1}{113400} \left (-4+k \right ) \left (k -3\right ) \left (-2+k \right ) \left (-1+k \right ) k x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 304
AsymptoticDSolveValue[2*x*y''[x]+(1+x)*y'[x]-k*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {4 \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {5}{4}-\frac {k}{2}\right ) \left (\frac {7}{4}-\frac {k}{2}\right ) \left (\frac {9}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^5}{155925}-\frac {2 \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {5}{4}-\frac {k}{2}\right ) \left (\frac {7}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^4}{2835}+\frac {4}{315} \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {5}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^3-\frac {2}{15} \left (\frac {3}{4}-\frac {k}{2}\right ) \left (\frac {k}{2}-\frac {1}{4}\right ) x^2+\frac {2}{3} \left (\frac {k}{2}-\frac {1}{4}\right ) x+1\right )+c_2 \left (\frac {2 \left (\frac {1}{2}-\frac {k}{2}\right ) \left (1-\frac {k}{2}\right ) \left (\frac {3}{2}-\frac {k}{2}\right ) \left (2-\frac {k}{2}\right ) k x^5}{14175}-\frac {1}{315} \left (\frac {1}{2}-\frac {k}{2}\right ) \left (1-\frac {k}{2}\right ) \left (\frac {3}{2}-\frac {k}{2}\right ) k x^4+\frac {2}{45} \left (\frac {1}{2}-\frac {k}{2}\right ) \left (1-\frac {k}{2}\right ) k x^3-\frac {1}{3} \left (\frac {1}{2}-\frac {k}{2}\right ) k x^2+k x+1\right ) \]