12.13.46 problem 45

Internal problem ID [1937]
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 : 45
Date solved : Monday, January 27, 2025 at 05:38:42 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

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

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

\[ y = 1-2 x +\frac {3}{2} x^{2}-\frac {11}{6} x^{3}+\frac {15}{8} x^{4}-\frac {71}{60} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 36

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

\[ y(x)\to -\frac {71 x^5}{60}+\frac {15 x^4}{8}-\frac {11 x^3}{6}+\frac {3 x^2}{2}-2 x+1 \]