12.13.9 problem 9

Internal problem ID [1900]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 7 Series Solutions of Linear Second Equations. 7.3 SERIES SOLUTIONS NEAR AN ORDINARY POINT II. Exercises 7.3. Page 338
Problem number : 9
Date solved : Monday, January 27, 2025 at 05:38:04 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (2 x^{2}+3 x \right ) y^{\prime \prime }+10 \left (1+x \right ) y^{\prime }+8 y&=0 \end{align*}

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

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

Order:=6; 
dsolve([(3*x+2*x^2)*diff(y(x),x$2)+10*(1+x)*diff(y(x),x)+8*y(x)=0,y(-1) = 1, D(y)(-1) = -1],y(x),type='series',x=-1);
 

\[ y = 1-\left (x +1\right )+4 \left (x +1\right )^{2}-\frac {13}{3} \left (x +1\right )^{3}+\frac {77}{6} \left (x +1\right )^{4}-\frac {278}{15} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 41

AsymptoticDSolveValue[{(3*x+2*x^2)*D[y[x],{x,2}]+10*(1+x)*D[y[x],x]+8*y[x]==0,{y[-1]==1,Derivative[1][y][-1]==-1}},y[x],{x,-1,"6"-1}]
 

\[ y(x)\to -\frac {278}{15} (x+1)^5+\frac {77}{6} (x+1)^4-\frac {13}{3} (x+1)^3+4 (x+1)^2-x \]