20.11.9 problem Problem 12

Internal problem ID [3791]
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 12
Date solved : Monday, January 27, 2025 at 08:02:21 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y&=8 x^{4} \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=x^{2} \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 19

dsolve([x^2*diff(y(x),x$2)-3*x*diff(y(x),x)+4*y(x)=8*x^4,x^2],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.019 (sec). Leaf size: 23

DSolve[x^2*D[y[x],{x,2}]-3*x*D[y[x],x]+4*y[x]==8*x^4,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to x^2 \left (2 x^2+2 c_2 \log (x)+c_1\right ) \]