62.29.6 problem Ex 7

Internal problem ID [12957]
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 : Tuesday, January 28, 2025 at 04:45:17 AM
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*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 34

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

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

✓ Solution by Mathematica

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

DSolve[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,y[x],x,IncludeSingularSolutions -> True]
 

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