83.41.23 problem 5 (x)

Internal problem ID [19503]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VIII. Linear equations of second order. Excercise at end of chapter VIII. Page 141
Problem number : 5 (x)
Date solved : Tuesday, January 28, 2025 at 01:46:57 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.012 (sec). Leaf size: 39

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

\[ y \left (x \right ) = x \,{\mathrm e}^{-x}+\sin \left (2 \left (x +1\right ) {\mathrm e}^{-x}\right ) c_{2} +\cos \left (2 \left (x +1\right ) {\mathrm e}^{-x}\right ) c_{1} +{\mathrm e}^{-x} \]

✓ Solution by Mathematica

Time used: 0.466 (sec). Leaf size: 47

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

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