12.13.41 problem 40

Internal problem ID [1932]
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 : 40
Date solved : Monday, January 27, 2025 at 05:38:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

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

✓ Solution by Maple

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

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

\[ y = -2+3 x +x^{2}-\frac {1}{6} x^{3}-\frac {3}{4} x^{4}+\frac {31}{120} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 32

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

\[ y(x)\to \frac {31 x^5}{120}-\frac {3 x^4}{4}-\frac {x^3}{6}+x^2+3 x-2 \]