56.29.6 problem Ex 7

Internal problem ID [14238]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VII, Linear differential equations with constant coefficients. Article 52. Summary. Page 117
Problem number : Ex 7
Date solved : Thursday, October 02, 2025 at 09:27:02 AM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

\begin{align*} y+3 x y^{\prime }+9 x^{2} y^{\prime \prime }+6 x^{3} y^{\prime \prime \prime }+x^{4} y^{\prime \prime \prime \prime }&=\left (1+\ln \left (x \right )\right )^{2} \end{align*}

✓ Maple. Time used: 0.093 (sec). Leaf size: 34

ode:=x^4*diff(diff(diff(diff(y(x),x),x),x),x)+6*x^3*diff(diff(diff(y(x),x),x),x)+9*x^2*diff(diff(y(x),x),x)+3*x*diff(y(x),x)+y(x) = (ln(x)+1)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (c_3 \ln \left (x \right )+c_1 \right ) \cos \left (\ln \left (x \right )\right )+\left (c_4 \ln \left (x \right )+c_2 \right ) \sin \left (\ln \left (x \right )\right )+\ln \left (x \right )^{2}+2 \ln \left (x \right )-3 \]

✓ Mathematica. Time used: 0.106 (sec). Leaf size: 39

ode=x^4*D[y[x],{x,4}]+6*x^3*D[y[x],{x,3}]+9*x^2*D[y[x],{x,2}]+3*x*D[y[x],x]+y[x]==(1+Log[x])^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \log ^2(x)+2 \log (x)+(c_2 \log (x)+c_1) \cos (\log (x))+(c_4 \log (x)+c_3) \sin (\log (x))-3 \end{align*}

✓ Sympy. Time used: 0.294 (sec). Leaf size: 48

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 4)) + 6*x**3*Derivative(y(x), (x, 3)) + 9*x**2*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) - (log(x) + 1)**2 + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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