12.12.24 problem 26

Internal problem ID [1878]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.2 SERIES SOLUTIONS NEAR AN ORDINARY POINT I. Exercises 7.2. Page 329
Problem number : 26
Date solved : Monday, January 27, 2025 at 05:37:43 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+4 x +5\right ) y^{\prime \prime }-20 \left (1+x \right ) y^{\prime }+60 y&=0 \end{align*}

Using series method with expansion around

\begin{align*} -1 \end{align*}

With initial conditions

\begin{align*} y \left (-1\right )&=3\\ y^{\prime }\left (-1\right )&=-3 \end{align*}

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 20

Order:=6; 
dsolve([(2*x^2+4*x+5)*diff(y(x),x$2)-20*(x+1)*diff(y(x),x)+60*y(x)=0,y(-1) = 3, D(y)(-1) = -3],y(x),type='series',x=-1);
 

\[ y = 3-3 \left (x +1\right )-30 \left (x +1\right )^{2}+\frac {20}{3} \left (x +1\right )^{3}+20 \left (x +1\right )^{4}-\frac {4}{3} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 42

AsymptoticDSolveValue[{(2*x^2+4*x+5)*D[y[x],{x,2}]-20*(x+1)*D[y[x],x]+60*y[x]==0,{y[-1]==3,Derivative[1][y][-1]==-3}},y[x],{x,-1,"6"-1}]
 

\[ y(x)\to -\frac {4}{3} (x+1)^5+20 (x+1)^4+\frac {20}{3} (x+1)^3-30 (x+1)^2-3 (x+1)+3 \]