15.11.34 problem 34

Internal problem ID [3144]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 19, page 86
Problem number : 34
Date solved : Sunday, March 30, 2025 at 01:19:34 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 8 y^{\prime \prime }-y&=x \,{\mathrm e}^{-\frac {x}{2}} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=3\\ y^{\prime }\left (0\right )&=5 \end{align*}

✓ Maple. Time used: 0.092 (sec). Leaf size: 34

ode:=8*diff(diff(y(x),x),x)-y(x) = x*exp(-1/2*x); 
ic:=y(0) = 3, D(y)(0) = 5; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \left (x +8\right ) {\mathrm e}^{-\frac {x}{2}}-5 \cosh \left (\frac {\sqrt {2}\, x}{4}\right )+16 \sqrt {2}\, \sinh \left (\frac {\sqrt {2}\, x}{4}\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 83

ode=8*D[y[x],{x,2}]-y[x]==x*Exp[-x/2]; 
ic={y[0]==3,Derivative[1][y][0] ==5}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{2} e^{-\frac {1}{4} \left (2+\sqrt {2}\right ) x} \left (2 e^{\frac {x}{2 \sqrt {2}}} (x+8)-\left (5+16 \sqrt {2}\right ) e^{x/2}+\left (16 \sqrt {2}-5\right ) e^{\frac {1}{2} \left (1+\sqrt {2}\right ) x}\right ) \]

✓ Sympy. Time used: 0.147 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*exp(-x/2) - y(x) + 8*Derivative(y(x), (x, 2)),0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): 5} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x e^{- \frac {x}{2}} + \left (- \frac {5}{2} + 8 \sqrt {2}\right ) e^{\frac {\sqrt {2} x}{4}} + \left (- 8 \sqrt {2} - \frac {5}{2}\right ) e^{- \frac {\sqrt {2} x}{4}} + 8 e^{- \frac {x}{2}} \]