12.16.14 problem 10

Internal problem ID [2076]
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 : 10
Date solved : Monday, January 27, 2025 at 05:41:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} \left (1+x \right ) y^{\prime \prime }-x \left (3+10 x \right ) y^{\prime }+30 y x&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 48

Order:=6; 
dsolve(x^2*(1+x)*diff(y(x),x$2)-x*(3+10*x)*diff(y(x),x)+30*x*y(x)=0,y(x),type='series',x=0);
 

\[ y = c_1 \,x^{4} \left (1-\frac {2}{5} x +\operatorname {O}\left (x^{6}\right )\right )+\left (43200 x^{4}-17280 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) \ln \left (x \right ) c_2 +\left (-144-1440 x -7200 x^{2}-28800 x^{3}-90720 x^{4}+82944 x^{5}+\operatorname {O}\left (x^{6}\right )\right ) c_2 \]

✓ Solution by Mathematica

Time used: 0.048 (sec). Leaf size: 48

AsymptoticDSolveValue[x^2*(1+x)*D[y[x],{x,2}]-x*(3+10*x)*D[y[x],x]+30*x*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_2 \left (x^4-\frac {2 x^5}{5}\right )+c_1 \left (745 x^4-300 x^4 \log (x)+200 x^3+50 x^2+10 x+1\right ) \]