77.26.19 problem 19

Internal problem ID [20628]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VI. Homogeneous linear equations with variable coefficients. Exercise VI (C) at page 93
Problem number : 19
Date solved : Thursday, October 02, 2025 at 06:17:39 PM
CAS classification : [[_3rd_order, _exact, _linear, _nonhomogeneous]]

\begin{align*} x^{3} y^{\prime \prime \prime }+2 x^{2} y^{\prime \prime }-x y^{\prime }+y&=x \ln \left (x \right ) \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 45

ode:=x^3*diff(diff(diff(y(x),x),x),x)+2*x^2*diff(diff(y(x),x),x)-x*diff(y(x),x)+y(x) = x*ln(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {2 \ln \left (x \right )^{3} x^{2}-3 \ln \left (x \right )^{2} x^{2}+24 x^{2} c_2 \ln \left (x \right )+24 c_3 \,x^{2}+6 c_1}{24 x} \]

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 45

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

\begin{align*} y(x)&\to \frac {1}{48} x \left (4 \log ^3(x)-6 \log ^2(x)+6 \log (x)-3\right )+\frac {c_1}{x}+c_2 x+c_3 x \log (x) \end{align*}

✓ Sympy. Time used: 0.216 (sec). Leaf size: 32

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

\[ y{\left (x \right )} = \frac {C_{1}}{x} + C_{2} x + C_{3} x \log {\left (x \right )} + \frac {x \log {\left (x \right )}^{3}}{12} - \frac {x \log {\left (x \right )}^{2}}{8} \]