7.16.14 problem 14

Internal problem ID [511]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.4 (Method of Frobenius: The exceptional cases). Problems at page 246
Problem number : 14
Date solved : Monday, January 27, 2025 at 02:54:16 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x \left (1+x \right ) y^{\prime }-4 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 45

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*(1+x)*diff(y(x),x)-4*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{2} \left (1-\frac {2}{5} x +\frac {1}{10} x^{2}-\frac {2}{105} x^{3}+\frac {1}{336} x^{4}-\frac {1}{2520} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\frac {c_2 \left (-144+96 x -24 x^{2}+2 x^{4}-\frac {4}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 63

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*(1+x)*D[y[x],x]-4*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {x^2}{72}+\frac {1}{x^2}-\frac {2}{3 x}+\frac {1}{6}\right )+c_2 \left (\frac {x^6}{336}-\frac {2 x^5}{105}+\frac {x^4}{10}-\frac {2 x^3}{5}+x^2\right ) \]