56.5.5 problem 5

Internal problem ID [8966]
Book : Own collection of miscellaneous problems
Section : section 5.0
Problem number : 5
Date solved : Monday, January 27, 2025 at 05:25:32 PM
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} \end{align*}

✓ Solution by Maple

Time used: 0.025 (sec). Leaf size: 21

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

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

✓ Solution by Mathematica

Time used: 0.045 (sec). Leaf size: 27

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

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