12.13.43 problem 42

Internal problem ID [1934]
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 : 42
Date solved : Monday, January 27, 2025 at 05:38:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

With initial conditions

\begin{align*} y \left (0\right )&=-3\\ y^{\prime }\left (0\right )&=5 \end{align*}

✓ Solution by Maple

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

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

\[ y = -3+5 x -5 x^{2}+\frac {23}{6} x^{3}-\frac {23}{12} x^{4}+\frac {11}{30} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 34

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

\[ y(x)\to \frac {11 x^5}{30}-\frac {23 x^4}{12}+\frac {23 x^3}{6}-5 x^2+5 x-3 \]