64.12.24 problem 24

Internal problem ID [13528]
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 : 24
Date solved : Tuesday, January 28, 2025 at 05:51:58 AM
CAS classification : [[_2nd_order, _exact, _linear, _nonhomogeneous]]

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 43

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

\[ y = \frac {4 x^{3}+\left (6 c_{1} +24 c_{2} +4\right ) x^{2}+\left (6 c_{1} +24 c_{2} +1\right ) x +6 c_{2}}{6+6 x} \]

✓ Solution by Mathematica

Time used: 0.644 (sec). Leaf size: 177

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

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