10.14.4 problem 4

Internal problem ID [1386]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number : 4
Date solved : Monday, January 27, 2025 at 04:56:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 24

Order:=6; 
dsolve([diff(y(x),x$2)+x^2*diff(y(x),x)+sin(x)*y(x)=0,y(0) = a__0, D(y)(0) = a__1],y(x),type='series',x=0);
 

\[ y = a_{0} +a_{1} x -\frac {1}{6} a_{0} x^{3}-\frac {1}{6} a_{1} x^{4}+\frac {1}{120} a_{0} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

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

AsymptoticDSolveValue[{D[y[x],{x,2}]+x^2*D[y[x],x]+Sin[x]*y[x]==0,{y[0]==a0,Derivative[1][y][0] ==a1}},y[x],{x,0,"6"-1}]
 

\[ y(x)\to \frac {\text {a0} x^5}{120}-\frac {\text {a0} x^3}{6}+\text {a0}-\frac {\text {a1} x^4}{6}+\text {a1} x \]