12.13.25 problem 28

Internal problem ID [1916]
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 : 28
Date solved : Monday, January 27, 2025 at 05:38:20 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (6+4 x \right ) y^{\prime \prime }+\left (2 x +1\right ) 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 )&=2 \end{align*}

✓ Solution by Maple

Time used: 0.001 (sec). Leaf size: 20

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

\[ y = -1+2 \left (x +1\right )-\frac {1}{4} \left (x +1\right )^{2}+\frac {1}{2} \left (x +1\right )^{3}-\frac {65}{96} \left (x +1\right )^{4}+\frac {67}{80} \left (x +1\right )^{5}+\operatorname {O}\left (\left (x +1\right )^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 46

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

\[ y(x)\to \frac {67}{80} (x+1)^5-\frac {65}{96} (x+1)^4+\frac {1}{2} (x+1)^3-\frac {1}{4} (x+1)^2+2 (x+1)-1 \]