54.5.4 problem 4

Internal problem ID [8619]
Book : Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition. 1997.
Section : CHAPTER 18. Power series solutions. 18.6. Indicial Equation with Equal Roots. Exercises page 373
Problem number : 4
Date solved : Monday, January 27, 2025 at 04:17:36 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+3 x y^{\prime }+\left (4 x^{2}+1\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 40

Order:=8; 
dsolve(x^2*diff(y(x),x$2)+3*x*diff(y(x),x)+(1+4*x^2)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {\left (c_{2} \ln \left (x \right )+c_{1} \right ) \left (1-x^{2}+\frac {1}{4} x^{4}-\frac {1}{36} x^{6}+\operatorname {O}\left (x^{8}\right )\right )+\left (x^{2}-\frac {3}{8} x^{4}+\frac {11}{216} x^{6}+\operatorname {O}\left (x^{8}\right )\right ) c_{2}}{x} \]

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 84

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+3*x*D[y[x],x]+(1+4*x^2)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to \frac {c_1 \left (-\frac {x^6}{36}+\frac {x^4}{4}-x^2+1\right )}{x}+c_2 \left (\frac {\frac {11 x^6}{216}-\frac {3 x^4}{8}+x^2}{x}+\frac {\left (-\frac {x^6}{36}+\frac {x^4}{4}-x^2+1\right ) \log (x)}{x}\right ) \]