62.29.6 problem Ex 7

Internal problem ID [12878]
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 : Wednesday, March 05, 2025 at 08:49:28 PM
CAS classification : [[_high_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.007 (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);
 

\[ y = \left (\ln \left (x \right ) c_3 +c_{1} \right ) \cos \left (\ln \left (x \right )\right )+\left (c_4 \ln \left (x \right )+c_{2} \right ) \sin \left (\ln \left (x \right )\right )+\ln \left (x \right )^{2}+2 \ln \left (x \right )-3 \]

✓ Mathematica. Time used: 0.212 (sec). Leaf size: 178

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]
 

\[ y(x)\to \log (x) \sin (\log (x)) \int _1^x-\frac {(\log (K[4])+1)^2 \sin (\log (K[4]))}{2 K[4]}dK[4]+\log (x) \cos (\log (x)) \int _1^x-\frac {\cos (\log (K[2])) (\log (K[2])+1)^2}{2 K[2]}dK[2]+\cos (\log (x)) \int _1^x\frac {(\log (K[1])+1)^2 (\cos (\log (K[1])) \log (K[1])-\sin (\log (K[1])))}{2 K[1]}dK[1]+\sin (\log (x)) \int _1^x\frac {(\log (K[3])+1)^2 (\cos (\log (K[3]))+\log (K[3]) \sin (\log (K[3])))}{2 K[3]}dK[3]+c_1 \cos (\log (x))+c_2 \log (x) \cos (\log (x))+c_3 \sin (\log (x))+c_4 \log (x) \sin (\log (x)) \]

✓ Sympy. Time used: 0.459 (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)
 

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