12.15.29 problem 25

Internal problem ID [2027]
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 : 25
Date solved : Monday, January 27, 2025 at 05:40:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.013 (sec). Leaf size: 54

Order:=8; 
dsolve(x^2*(1+4*x)*diff(y(x),x$2)-x*(1-4*x)*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
 

\[ y = x \left (\left (c_2 \ln \left (x \right )+c_1 \right ) \left (1-5 x +\frac {85}{4} x^{2}-\frac {3145}{36} x^{3}+\frac {204425}{576} x^{4}-\frac {825877}{576} x^{5}+\frac {119752165}{20736} x^{6}-\frac {23591176505}{1016064} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+\left (2 x -\frac {39}{4} x^{2}+\frac {4499}{108} x^{3}-\frac {594305}{3456} x^{4}+\frac {2420617}{3456} x^{5}-\frac {117547073}{41472} x^{6}+\frac {162576422327}{14224896} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.016 (sec). Leaf size: 158

AsymptoticDSolveValue[x^2*(1+4*x)*D[y[x],{x,2}]-x*(1-4*x)*D[y[x],x]+(1+x)*y[x]==0,y[x],{x,0,"8"-1}]
 

\[ y(x)\to c_1 x \left (-\frac {23591176505 x^7}{1016064}+\frac {119752165 x^6}{20736}-\frac {825877 x^5}{576}+\frac {204425 x^4}{576}-\frac {3145 x^3}{36}+\frac {85 x^2}{4}-5 x+1\right )+c_2 \left (x \left (\frac {162576422327 x^7}{14224896}-\frac {117547073 x^6}{41472}+\frac {2420617 x^5}{3456}-\frac {594305 x^4}{3456}+\frac {4499 x^3}{108}-\frac {39 x^2}{4}+2 x\right )+x \left (-\frac {23591176505 x^7}{1016064}+\frac {119752165 x^6}{20736}-\frac {825877 x^5}{576}+\frac {204425 x^4}{576}-\frac {3145 x^3}{36}+\frac {85 x^2}{4}-5 x+1\right ) \log (x)\right ) \]