64.13.17 problem 17

Internal problem ID [13547]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.5. The Cauchy-Euler Equation. Exercises page 169
Problem number : 17
Date solved : Tuesday, January 28, 2025 at 05:52:28 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 27

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

\[ y = \sin \left (2 \ln \left (x \right )\right ) c_{2} +\cos \left (2 \ln \left (x \right )\right ) c_{1} +\frac {2 \ln \left (x \right ) x}{5}-\frac {4 x}{25} \]

✓ Solution by Mathematica

Time used: 0.076 (sec). Leaf size: 69

DSolve[x^2*D[y[x],{x,2}]+x*D[y[x],x]+4*y[x]==2*x*Log[x],y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \cos (2 \log (x)) \int _1^x-\log (K[1]) \sin (2 \log (K[1]))dK[1]+\sin (2 \log (x)) \int _1^x\cos (2 \log (K[2])) \log (K[2])dK[2]+c_1 \cos (2 \log (x))+c_2 \sin (2 \log (x)) \]