12.9.14 problem 14

Internal problem ID [1770]
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 : 14
Date solved : Monday, January 27, 2025 at 05:34:39 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{-x} \end{align*}

✓ Solution by Maple

Time used: 0.010 (sec). Leaf size: 19

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

\[ y = {\mathrm e}^{-x} \left (c_2 +c_1 \sqrt {x}+x^{{3}/{2}}\right ) \]

✓ Solution by Mathematica

Time used: 0.047 (sec). Leaf size: 28

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

\[ y(x)\to e^{-x} \left (x^{3/2}+2 c_2 \sqrt {x}+c_1\right ) \]