83.36.9 problem 9

Internal problem ID [19443]
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 VIII (A) at page 125
Problem number : 9
Date solved : Tuesday, January 28, 2025 at 01:43:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 14

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

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

✓ Solution by Mathematica

Time used: 0.028 (sec). Leaf size: 19

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

\[ y(x)\to x \left (e^x (x-1+c_2)+c_1\right ) \]