12.13.35 problem 35

Internal problem ID [1926]
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 : 35
Date solved : Monday, January 27, 2025 at 05:38:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 x y^{\prime }-\left (3 x^{2}+2\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 )&=-5 \end{align*}

✓ Solution by Maple

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

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

\[ y = 2-5 x +2 x^{2}-\frac {10}{3} x^{3}+\frac {3}{2} x^{4}-\frac {25}{12} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

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

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

\[ y(x)\to -\frac {25 x^5}{12}+\frac {3 x^4}{2}-\frac {10 x^3}{3}+2 x^2-5 x+2 \]