Internal problem ID [12275]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {y^{\prime \prime }+\frac {2 x y^{\prime }}{2 x -1}-\frac {4 x y}{\left (2 x -1\right )^{2}}=0} \]
✓ Solution by Maple
Time used: 0.094 (sec). Leaf size: 77
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)
\[ y \left (x \right ) = -\frac {2 \,2^{\frac {3}{4}} {\mathrm e}^{-\frac {1}{4}} \left (\frac {\left (\frac {2 \left (\operatorname {erf}\left (\frac {\sqrt {-2+4 x}}{2}\right )-1\right ) c_{2}}{3}+\operatorname {erf}\left (\frac {\sqrt {-2+4 x}}{2}\right ) c_{1} \right ) \sqrt {-2+4 x}\, \left (-\frac {1}{2}+x \right ) \sqrt {\pi }}{2}+\left (-1+x \right ) {\mathrm e}^{\frac {1}{2}-x} \left (c_{1} +\frac {2 c_{2}}{3}\right )\right )}{\sqrt {-2+4 x}} \]
✓ Solution by Mathematica
Time used: 0.508 (sec). Leaf size: 64
DSolve[y''[x]+ (2*x)/(2*x-1)*y'[x]- 4*x/( (2*x-1)^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 (2 x-1)+\frac {1}{6} c_2 \left (\frac {4 e^{\frac {1}{2}-x} (x-1)}{\sqrt {2 x-1}}+\sqrt {2} (1-2 x) \Gamma \left (\frac {1}{2},x-\frac {1}{2}\right )\right ) \]