45.3.8 problem 8

Internal problem ID [7266]
Book : A FIRST COURSE IN DIFFERENTIAL EQUATIONS with Modeling Applications. Dennis G. Zill. 9th edition. Brooks/Cole. CA, USA.
Section : Chapter 6. SERIES SOLUTIONS OF LINEAR EQUATIONS. Exercises. 6.3.1 page 250
Problem number : 8
Date solved : Monday, January 27, 2025 at 02:49:18 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (36 x^{2}-\frac {1}{4}\right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 34

Order:=6; 
dsolve(x^2*diff(y(x),x$2)+x*diff(y(x),x)+(36*x^2-1/4)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_{1} x \left (1-6 x^{2}+\frac {54}{5} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (1-18 x^{2}+54 x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 52

AsymptoticDSolveValue[x^2*D[y[x],{x,2}]+x*D[y[x],x]+(36*x^2-1/4)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (54 x^{7/2}-18 x^{3/2}+\frac {1}{\sqrt {x}}\right )+c_2 \left (\frac {54 x^{9/2}}{5}-6 x^{5/2}+\sqrt {x}\right ) \]