85.67.20 problem 20

Internal problem ID [22911]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 4. Linear differential equations. A Exercises at page 216
Problem number : 20
Date solved : Thursday, October 02, 2025 at 09:16:31 PM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 36

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

\[ y = \frac {x^{2} c_1 \ln \left (x \right )}{2}-\frac {c_1 \,x^{2}}{4}+\frac {c_2 \,x^{2}}{2}+\frac {\ln \left (x \right )^{2}}{8}+\frac {\ln \left (x \right )}{4}+c_3 \]

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 43

ode=x^2*D[y[x],{x,3}]-x*D[y[x],{x,2}]+D[y[x],x]==Log[x]/x; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{8} \left ((4 c_1-2 c_2) x^2+\left (2+4 c_2 x^2\right ) \log (x)+\log ^2(x)+8 c_3\right ) \end{align*}

✓ Sympy. Time used: 0.185 (sec). Leaf size: 29

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

\[ y{\left (x \right )} = C_{1} + C_{2} x^{2} + C_{3} x^{2} \log {\left (x \right )} + \frac {\log {\left (x \right )}^{2}}{8} + \frac {\log {\left (x \right )}}{4} \]