67.2.39 problem Problem 18(d)

Internal problem ID [13996]
Book : APPLIED DIFFERENTIAL EQUATIONS The Primary Course by Vladimir A. Dobrushkin. CRC Press 2015
Section : Chapter 4, Second and Higher Order Linear Differential Equations. Problems page 221
Problem number : Problem 18(d)
Date solved : Tuesday, January 28, 2025 at 06:11:17 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 38

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

\[ y = \frac {\left (c_{1} +2\right ) \left (x -1\right ) \ln \left (x -1\right )}{4}-\frac {\left (c_{1} -2\right ) \left (x -1\right ) \ln \left (x +1\right )}{4}+c_{2} x +\frac {c_{1}}{2}-c_{2} \]

✓ Solution by Mathematica

Time used: 0.664 (sec). Leaf size: 235

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

\[ y(x)\to \frac {\exp \left (\int _1^x\frac {3 K[1]+1}{2 \left (K[1]^2-1\right )}dK[1]\right ) \left (\int _1^x-\frac {\exp \left (\int _1^{K[3]}\frac {3 K[1]+1}{2 \left (K[1]^2-1\right )}dK[1]\right ) (2 K[3]-1) \int _1^{K[3]}\exp \left (-2 \int _1^{K[2]}\frac {3 K[1]+1}{2 \left (K[1]^2-1\right )}dK[1]\right )dK[2]}{(K[3]-1) \sqrt {K[3]+1}}dK[3]+\int _1^x\exp \left (-2 \int _1^{K[2]}\frac {3 K[1]+1}{2 \left (K[1]^2-1\right )}dK[1]\right )dK[2] \left (\int _1^x\frac {\exp \left (\int _1^{K[4]}\frac {3 K[1]+1}{2 \left (K[1]^2-1\right )}dK[1]\right ) (2 K[4]-1)}{(K[4]-1) \sqrt {K[4]+1}}dK[4]+c_2\right )+c_1\right )}{\sqrt {x+1}} \]