12.16.7 problem 3

Internal problem ID [2069]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS III. Exercises 7.7. Page 389
Problem number : 3
Date solved : Monday, January 27, 2025 at 05:41:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x \left (2 x +1\right ) y^{\prime }-\left (1+3 x \right ) y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 46

Order:=6; 
dsolve(4*x^2*(1+x)*diff(y(x),x$2)+4*x*(1+2*x)*diff(y(x),x)-(1+3*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \frac {c_1 x \left (1+\operatorname {O}\left (x^{6}\right )\right )+\left (x +\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +\left (1-x -x^{2}+\frac {1}{2} x^{3}-\frac {1}{3} x^{4}+\frac {1}{4} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 53

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

\[ y(x)\to c_1 \left (\sqrt {x} \log (x)-\frac {2 x^4-3 x^3+6 x^2+6 x-6}{6 \sqrt {x}}\right )+c_2 \sqrt {x} \]