73.9.30 problem 14.3 (f)

Internal problem ID [15291]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 14. Higher order equations and the reduction of order method. Additional exercises page 277
Problem number : 14.3 (f)
Date solved : Tuesday, January 28, 2025 at 07:51:39 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&={\mathrm e}^{-x} \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 18

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

\[ y = c_{2} x +c_{1} {\mathrm e}^{-x}+x^{2}+1 \]

✓ Solution by Mathematica

Time used: 0.239 (sec). Leaf size: 241

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

\[ y(x)\to \exp \left (\int _1^x-\frac {K[1]+2}{2 K[1]+2}dK[1]-\frac {1}{2} \int _1^x\frac {K[2]}{K[2]+1}dK[2]\right ) \left (\int _1^x-\exp \left (\int _1^{K[4]}-\frac {K[1]+2}{2 K[1]+2}dK[1]+\frac {1}{2} \int _1^{K[4]}\frac {K[2]}{K[2]+1}dK[2]\right ) (K[4]+1) \int _1^{K[4]}\exp \left (-2 \int _1^{K[3]}-\frac {K[1]+2}{2 K[1]+2}dK[1]\right )dK[3]dK[4]+\int _1^x\exp \left (-2 \int _1^{K[3]}-\frac {K[1]+2}{2 K[1]+2}dK[1]\right )dK[3] \left (\int _1^x\exp \left (\int _1^{K[5]}-\frac {K[1]+2}{2 K[1]+2}dK[1]+\frac {1}{2} \int _1^{K[5]}\frac {K[2]}{K[2]+1}dK[2]\right ) (K[5]+1)dK[5]+c_2\right )+c_1\right ) \]