12.9.27 problem 27

Internal problem ID [1783]
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 : 27
Date solved : Monday, January 27, 2025 at 05:34:48 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} 4 x^{2} y^{\prime \prime }-4 x y^{\prime }+\left (-16 x^{2}+3\right ) y&=0 \end{align*}

Using reduction of order method given that one solution is

\begin{align*} y&=\sqrt {x}\, {\mathrm e}^{2 x} \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 21

dsolve([4*x^2*diff(y(x),x$2)-4*x*diff(y(x),x)+(3-16*x^2)*y(x)=0,sqrt(x)*exp(2*x)],singsol=all)
 

\[ y = \sqrt {x}\, \left (c_1 \sinh \left (2 x \right )+c_2 \cosh \left (2 x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.033 (sec). Leaf size: 32

DSolve[4*x^2*D[y[x],{x,2}]-4*x*D[y[x],x]+(3-16*x^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to \frac {1}{4} e^{-2 x} \sqrt {x} \left (c_2 e^{4 x}+4 c_1\right ) \]