34.12.3 problem 8

Internal problem ID [8038]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 17. Linear equations with variable coefficients (Cauchy and Legndre). Supplemetary problems. Page 110
Problem number : 8
Date solved : Tuesday, September 30, 2025 at 05:14:34 PM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 40

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

\[ y = -c_1 \ln \left (x \right )+\ln \left (x \right ) x +c_2 x +c_3 -x +\frac {\tan \left (\frac {\ln \left (x \right )}{2}\right )+1}{1+\tan \left (\frac {\ln \left (x \right )}{2}\right )^{2}} \]

✓ Mathematica. Time used: 32.715 (sec). Leaf size: 54

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

\begin{align*} y(x)&\to \int _1^x\int _1^{K[3]}\frac {c_1+\int _1^{K[2]}\left (\frac {\sin (\log (K[1]))}{K[1]}+1\right )dK[1]}{K[2]^2}dK[2]dK[3]+c_3 x+c_2 \end{align*}

✓ Sympy. Time used: 0.193 (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 - sin(log(x)),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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