12.15.51 problem 47

Internal problem ID [2049]
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 II. Exercises 7.6. Page 374
Problem number : 47
Date solved : Monday, January 27, 2025 at 05:41:05 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} \left (1+x \right ) y^{\prime \prime }+4 x^{2} y^{\prime }+\left (1-5 x \right ) 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: 48

Order:=6; 
dsolve(4*x^2*(1+x)*diff(y(x),x$2)+4*x^2*diff(y(x),x)+(1-5*x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = \sqrt {x}\, \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1+x -\frac {1}{4} x^{2}+\frac {5}{36} x^{3}-\frac {55}{576} x^{4}+\frac {209}{2880} x^{5}+\operatorname {O}\left (x^{6}\right )\right )+\left (\left (-3\right ) x +\frac {1}{4} x^{2}-\frac {5}{54} x^{3}+\frac {175}{3456} x^{4}-\frac {2863}{86400} x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.008 (sec). Leaf size: 124

AsymptoticDSolveValue[4*x^2*(1+x)*D[y[x],{x,2}]+4*x^2*D[y[x],x]+(1-5*x)*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \sqrt {x} \left (\frac {209 x^5}{2880}-\frac {55 x^4}{576}+\frac {5 x^3}{36}-\frac {x^2}{4}+x+1\right )+c_2 \left (\sqrt {x} \left (-\frac {2863 x^5}{86400}+\frac {175 x^4}{3456}-\frac {5 x^3}{54}+\frac {x^2}{4}-3 x\right )+\sqrt {x} \left (\frac {209 x^5}{2880}-\frac {55 x^4}{576}+\frac {5 x^3}{36}-\frac {x^2}{4}+x+1\right ) \log (x)\right ) \]