20.11.12 problem Problem 15

Internal problem ID [3794]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.9, Reduction of Order. page 572
Problem number : Problem 15
Date solved : Monday, January 27, 2025 at 08:02:24 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\sqrt {x} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 20

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

\[ y \left (x \right ) = \left (c_{2} +\ln \left (x \right ) c_{1} +\frac {\ln \left (x \right )^{3}}{24}\right ) \sqrt {x} \]

✓ Solution by Mathematica

Time used: 0.025 (sec). Leaf size: 29

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

\[ y(x)\to \frac {1}{24} \sqrt {x} \left (\log ^3(x)+12 c_2 \log (x)+24 c_1\right ) \]