7.15.5 problem 5

Internal problem ID [461]
Book : Elementary Differential Equations. By C. Henry Edwards, David E. Penney and David Calvis. 6th edition. 2008
Section : Chapter 3. Power series methods. Section 3.3 (Regular singular points). Problems at page 231
Problem number : 5
Date solved : Monday, January 27, 2025 at 02:53:53 AM
CAS classification : [[_Emden, _Fowler]]

\begin{align*} x \left (1+x \right ) y^{\prime \prime }+2 y^{\prime }+3 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 60

Order:=6; 
dsolve(x*(1+x)*diff(y(x),x$2)+2*diff(y(x),x)+3*x*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 \left (1-\frac {1}{2} x^{2}+\frac {1}{12} x^{3}+\frac {1}{20} x^{4}-\frac {17}{600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) x +\left (\left (-2\right ) x +x^{3}-\frac {1}{6} x^{4}-\frac {1}{10} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +\left (1+2 x -\frac {5}{2} x^{2}-\frac {11}{6} x^{3}+\frac {7}{9} x^{4}+\frac {77}{600} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{x} \]

✓ Solution by Mathematica

Time used: 0.035 (sec). Leaf size: 77

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

\[ y(x)\to c_2 \left (\frac {x^4}{20}+\frac {x^3}{12}-\frac {x^2}{2}+1\right )+c_1 \left (\frac {37 x^4-120 x^3-90 x^2+180 x+36}{36 x}-\frac {1}{6} \left (x^3-6 x^2+12\right ) \log (x)\right ) \]