36.5.4 problem 4

Internal problem ID [6348]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 4
Date solved : Monday, January 27, 2025 at 01:57:49 PM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} \left (x^{2}+x \right ) y^{\prime \prime }+3 y^{\prime }-6 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: 60

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

\[ y = \frac {c_1 \left (1+\frac {3}{4} x^{2}-\frac {1}{10} x^{3}+\frac {17}{80} x^{4}-\frac {9}{100} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x^{2}+c_2 \left (\ln \left (x \right ) \left (6 x^{2}+\frac {9}{2} x^{4}-\frac {3}{5} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (-2-12 x -24 x^{2}-22 x^{3}-\frac {171}{8} x^{4}-\frac {653}{100} x^{5}+\operatorname {O}\left (x^{6}\right )\right )\right )}{x^{2}} \]

✓ Solution by Mathematica

Time used: 0.040 (sec). Leaf size: 73

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

\[ y(x)\to c_2 \left (\frac {7 x^4}{20}-\frac {x^3}{6}+x^2+1\right )+c_1 \left (\frac {1}{3} \left (x^3-6 x^2-6\right ) \log (x)+\frac {7 x^4+240 x^3+72 x^2+180 x+36}{36 x}\right ) \]