54.9.15 problem 15

Internal problem ID [8685]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number : 15
Date solved : Monday, January 27, 2025 at 04:19:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-x^{2} y^{\prime }+y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 54

Order:=8; 
dsolve(4*x^2*diff(y(x),x$2)-x^2*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
 

\[ y = \left (\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1+\frac {1}{8} x +\frac {3}{256} x^{2}+\frac {5}{6144} x^{3}+\frac {35}{786432} x^{4}+\frac {21}{10485760} x^{5}+\frac {77}{1006632960} x^{6}+\frac {143}{56371445760} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (-\frac {1}{256} x^{2}-\frac {1}{2048} x^{3}-\frac {19}{524288} x^{4}-\frac {25}{12582912} x^{5}-\frac {317}{3623878656} x^{6}-\frac {469}{144955146240} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_{2} \right ) \sqrt {x} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 171

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]-x^2*D[y[x],x]+y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {143 x^7}{56371445760}+\frac {77 x^6}{1006632960}+\frac {21 x^5}{10485760}+\frac {35 x^4}{786432}+\frac {5 x^3}{6144}+\frac {3 x^2}{256}+\frac {x}{8}+1\right )+c_2 \left (\sqrt {x} \left (-\frac {469 x^7}{144955146240}-\frac {317 x^6}{3623878656}-\frac {25 x^5}{12582912}-\frac {19 x^4}{524288}-\frac {x^3}{2048}-\frac {x^2}{256}\right )+\sqrt {x} \left (\frac {143 x^7}{56371445760}+\frac {77 x^6}{1006632960}+\frac {21 x^5}{10485760}+\frac {35 x^4}{786432}+\frac {5 x^3}{6144}+\frac {3 x^2}{256}+\frac {x}{8}+1\right ) \log (x)\right ) \]