45.3.12 problem 14

Internal problem ID [7270]
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 : 14
Date solved : Monday, January 27, 2025 at 02:49:24 PM
CAS classification : [_Lienard]

\begin{align*} x y^{\prime \prime }+3 y^{\prime }+y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 46

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

\[ y = \frac {c_{1} x^{2} \left (1-\frac {1}{8} x^{2}+\frac {1}{192} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \left (x \right ) \left (x^{2}-\frac {1}{8} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2+\frac {3}{32} x^{4}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 57

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

\[ y(x)\to c_2 \left (\frac {x^4}{192}-\frac {x^2}{8}+1\right )+c_1 \left (\frac {1}{16} \left (x^2-8\right ) \log (x)-\frac {5 x^4-16 x^2-64}{64 x^2}\right ) \]