45.2.5 problem 5

Internal problem ID [7228]
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. 6.2 page 239
Problem number : 5
Date solved : Monday, January 27, 2025 at 02:48:32 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{3}+4 x \right ) y^{\prime \prime }-2 x y^{\prime }+6 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.009 (sec). Leaf size: 60

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

\[ y = c_{1} x \left (1-\frac {1}{2} x +\frac {1}{24} x^{2}+\frac {1}{48} x^{3}-\frac {1}{384} x^{4}-\frac {5}{2304} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (-\frac {3}{2} x +\frac {3}{4} x^{2}-\frac {1}{16} x^{3}-\frac {1}{32} x^{4}+\frac {1}{256} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (1+\frac {1}{2} x -\frac {7}{4} x^{2}+\frac {31}{96} x^{3}+\frac {1}{24} x^{4}-\frac {67}{3072} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.041 (sec). Leaf size: 85

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

\[ y(x)\to c_1 \left (\frac {1}{96} \left (7 x^4+37 x^3-240 x^2+192 x+96\right )-\frac {1}{32} x \left (x^3+2 x^2-24 x+48\right ) \log (x)\right )+c_2 \left (-\frac {x^5}{384}+\frac {x^4}{48}+\frac {x^3}{24}-\frac {x^2}{2}+x\right ) \]