62.30.2 problem Ex 2

Internal problem ID [12966]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 53. Change of dependent variable. Page 125
Problem number : Ex 2
Date solved : Tuesday, January 28, 2025 at 04:45:37 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime }-\left (2 x +1\right ) y^{\prime }+\left (1+x \right ) y&=x^{2}-x -1 \end{align*}

✓ Solution by Maple

Time used: 0.005 (sec). Leaf size: 16

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

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

✓ Solution by Mathematica

Time used: 0.266 (sec). Leaf size: 62

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

\[ y(x)\to \frac {1}{2} \left (2 e^x \int _1^x\frac {1}{2} e^{-K[1]} \left (-K[1]^2+K[1]+1\right )dK[1]+x^2 \left (-1+c_2 e^x\right )+x+2 c_1 e^x\right ) \]