84.30.7 problem 18.17

Internal problem ID [22296]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 18. Linear differential equations with variable coefficients. Supplementary problems
Problem number : 18.17
Date solved : Thursday, October 02, 2025 at 08:37:13 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Maple. Time used: 0.004 (sec). Leaf size: 39

Order:=6; 
ode:=exp(x)*diff(diff(y(x),x),x)+sin(x)*diff(y(x),x)+x*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \left (1-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}\right ) y \left (0\right )+\left (x -\frac {1}{6} x^{3}+\frac {7}{120} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Mathematica. Time used: 0.001 (sec). Leaf size: 42

ode=Exp[x]*D[y[x],{x,2}]+Sin[x]*D[y[x],x]+x*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_2 \left (\frac {7 x^5}{120}-\frac {x^3}{6}+x\right )+c_1 \left (\frac {x^4}{12}-\frac {x^3}{6}+1\right ) \]

✓ Sympy. Time used: 0.843 (sec). Leaf size: 85

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x*y(x) + exp(x)*Derivative(y(x), (x, 2)) + sin(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_ordinary",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} \left (\frac {x^{4} e^{- 2 x} \sin {\left (x \right )}}{24} - \frac {x^{3} e^{- x}}{6} + 1\right ) + C_{1} x \left (- \frac {x^{3} e^{- x}}{12} - \frac {x^{3} e^{- 3 x} \sin ^{3}{\left (x \right )}}{24} + \frac {x^{2} e^{- 2 x} \sin ^{2}{\left (x \right )}}{6} - \frac {x e^{- x} \sin {\left (x \right )}}{2} + 1\right ) + O\left (x^{6}\right ) \]