45.3.24 problem 26

Internal problem ID [7282]
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 : 26
Date solved : Monday, January 27, 2025 at 02:49:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (16 x^{4}+3\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: 30

Order:=6; 
dsolve(4*x^2*diff(y(x),x$2)-4*x*diff(y(x),x)+(16*x^4+3)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (x \left (1-\frac {1}{5} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{1} +\left (1-\frac {1}{3} x^{4}+\operatorname {O}\left (x^{6}\right )\right ) c_{2} \right ) \]

✓ Solution by Mathematica

Time used: 0.010 (sec). Leaf size: 40

AsymptoticDSolveValue[4*x^2*D[y[x],{x,2}]-4*x*D[y[x],x]+(16*x^4+3)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\sqrt {x}-\frac {x^{9/2}}{3}\right )+c_2 \left (x^{3/2}-\frac {x^{11/2}}{5}\right ) \]