64.12.21 problem 21

Internal problem ID [13525]
Book : Differential Equations by Shepley L. Ross. Third edition. John Willey. New Delhi. 2004.
Section : Chapter 4, Section 4.4. Variation of parameters. Exercises page 162
Problem number : 21
Date solved : Tuesday, January 28, 2025 at 05:51:52 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 24

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

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

✓ Solution by Mathematica

Time used: 0.232 (sec). Leaf size: 266

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

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