74.16.17 problem 19

Internal problem ID [16523]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.8, page 203
Problem number : 19
Date solved : Tuesday, January 28, 2025 at 09:10:10 AM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Solution by Maple

Time used: 0.002 (sec). Leaf size: 14

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

\[ y = 1+\frac {1}{6} x^{3}-\frac {1}{30} x^{5}+\operatorname {O}\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 19

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

\[ y(x)\to -\frac {7 x^5}{120}+\frac {x^3}{6}+1 \]