67.2.54 problem Problem 20(a)

Internal problem ID [14011]
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 20(a)
Date solved : Tuesday, January 28, 2025 at 06:12:31 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }+\frac {2 x y^{\prime }}{2 x -1}-\frac {4 x y}{\left (2 x -1\right )^{2}}&=0 \end{align*}

✓ Solution by Maple

Time used: 0.231 (sec). Leaf size: 86

dsolve(diff(y(x),x$2)+ (2*x)/(2*x-1)*diff(y(x),x)- 4*x/( (2*x-1)^2)*y(x)=0,y(x), singsol=all)
 

\[ y = -\frac {\left (\sqrt {\pi }\, \sqrt {2}\, \left (c_{1} +\frac {2 c_{2}}{3}\right ) \left (-\frac {1}{2}+x \right ) \sqrt {-1+2 x}\, \operatorname {erf}\left (\frac {\sqrt {4 x -2}}{2}\right )-\frac {2 c_{2} \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {1}{2}+x \right ) \sqrt {-1+2 x}}{3}+2 \,{\mathrm e}^{\frac {1}{2}-x} \left (c_{1} +\frac {2 c_{2}}{3}\right ) \left (x -1\right )\right ) {\mathrm e}^{-\frac {1}{4}} 2^{{1}/{4}}}{\sqrt {-1+2 x}} \]

✓ Solution by Mathematica

Time used: 0.190 (sec). Leaf size: 88

DSolve[D[y[x],{x,2}]+ (2*x)/(2*x-1)*D[y[x],x]- 4*x/( (2*x-1)^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
 

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