12.13.32 problem 32

Internal problem ID [1923]
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 : 32
Date solved : Monday, January 27, 2025 at 05:38:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+2 x y^{\prime }+\left (2 x^{2}+3\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: 0.000 (sec). Leaf size: 20

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

\[ y = 1-2 x -\frac {3}{2} x^{2}+\frac {5}{3} x^{3}+\frac {17}{24} x^{4}-\frac {11}{20} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

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

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

\[ y(x)\to -\frac {11 x^5}{20}+\frac {17 x^4}{24}+\frac {5 x^3}{3}-\frac {3 x^2}{2}-2 x+1 \]