12.9.6 problem 6

Internal problem ID [1762]
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 : 6
Date solved : Monday, January 27, 2025 at 05:34:32 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} 4 x^{2} y^{\prime \prime }+\left (-8 x^{2}+4 x \right ) y^{\prime }+\left (4 x^{2}-4 x -1\right ) y&=4 \sqrt {x}\, {\mathrm e}^{x} \left (1+4 x \right ) \end{align*}

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.019 (sec). Leaf size: 26

dsolve([4*x^2*diff(y(x),x$2)+(4*x-8*x^2)*diff(y(x),x)+(4*x^2-4*x-1)*y(x)=4*x^(1/2)*exp(x)*(1+4*x),x^(1/2)*exp(x)],singsol=all)
 

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

✓ Solution by Mathematica

Time used: 0.043 (sec). Leaf size: 32

DSolve[4*x^2*D[y[x],{x,2}]+(4*x-8*x^2)*D[y[x],x]+(4*x^2-4*x-1)*y[x]==4*x^(1/2)*Exp[x]*(1+4*x),y[x],x,IncludeSingularSolutions -> True]
 

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