15.12.24 problem 24

Internal problem ID [3168]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 20, page 90
Problem number : 24
Date solved : Tuesday, March 04, 2025 at 04:04:35 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 y^{\prime \prime }-4 y^{\prime }+y&={\mathrm e}^{\frac {x}{2}} \ln \left (x \right ) \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 29

ode:=4*diff(diff(y(x),x),x)-4*diff(y(x),x)+y(x) = exp(1/2*x)*ln(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (\ln \left (x \right ) x^{2}-\frac {3 x^{2}}{2}+8 c_{1} x +8 c_2 \right ) {\mathrm e}^{\frac {x}{2}}}{8} \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 38

ode=4*D[y[x],{x,2}]-4*D[y[x],x]+y[x]==Exp[x/2]*Log[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \frac {1}{16} e^{x/2} \left (-3 x^2+2 x^2 \log (x)+16 c_2 x+16 c_1\right ) \]

✓ Sympy. Time used: 0.363 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x) - exp(x/2)*log(x) - 4*Derivative(y(x), x) + 4*Derivative(y(x), (x, 2)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (C_{1} + x \left (C_{2} + \frac {x \log {\left (x \right )}}{8} - \frac {3 x}{16}\right )\right ) e^{\frac {x}{2}} \]