12.9.13 problem 13

Internal problem ID [1769]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 13
Date solved : Monday, January 27, 2025 at 05:34:38 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }-3 x y^{\prime }+4 y&=4 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: 17

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

\[ y = x^{2} \left (\ln \left (x \right ) c_1 +x^{2}+c_2 \right ) \]

✓ Solution by Mathematica

Time used: 0.021 (sec). Leaf size: 21

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

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