12.9.16 problem 16

Internal problem ID [1772]
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 : 16
Date solved : Monday, January 27, 2025 at 05:34:40 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 x^{2} y^{\prime \prime }-4 x \left (1+x \right ) y^{\prime }+\left (2 x +3\right ) y&=4 x^{{5}/{2}} {\mathrm e}^{2 x} \end{align*}

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.011 (sec). Leaf size: 20

dsolve([4*x^2*diff(y(x),x$2)-4*x*(x+1)*diff(y(x),x)+(2*x+3)*y(x)=4*x^(5/2)*exp(2*x),x^(1/2)],singsol=all)
 

\[ y = \left (c_2 +{\mathrm e}^{x} c_1 +\frac {{\mathrm e}^{2 x}}{2}\right ) \sqrt {x} \]

✓ Solution by Mathematica

Time used: 0.036 (sec). Leaf size: 31

DSolve[4*x^2*D[y[x],{x,2}]-4*x*(x+1)*D[y[x],x]+(2*x+3)*y[x]==4*x^(5/2)*Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
 

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