Mister Exam

# Differential equation xy''-(2x+1)y'+(x+1)y=0

y() =
y'() =
y''() =
y'''() =
y''''() =

from to

### The solution

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    2
d                                   d
x*---(y(x)) + (1 + x)*y(x) - (1 + 2*x)*--(y(x)) = 0
2                                  dx
dx                                               
$$x \frac{d^{2}}{d x^{2}} y{\left(x \right)} + \left(x + 1\right) y{\left(x \right)} - \left(2 x + 1\right) \frac{d}{d x} y{\left(x \right)} = 0$$
x*y'' + (x + 1)*y - (2*x + 1)*y' = 0
$$y\left(x\right)={\it ilt}\left(-{{\left(g_{19164}^2-2\,g_{19164}+1 \right)\,\left({{d}\over{d\,g_{19164}}}\,\mathcal{L}\left(y\left(x \right) , x , g_{19164}\right)\right)-2\,y\left(0\right)}\over{3\, g_{19164}-3}} , g_{19164} , x\right)$$
y = 'ilt(-((g19164^2-2*g19164+1)*'diff('laplace(y,x,g19164),g19164,1)-2*y(0))/(3*g19164-3),g19164,x)
          /           4      5    3\         /         2    3\
\         24     40    3 /         \        2    6 /        
$$y{\left(x \right)} = C_{2} \left(- \frac{3 x^{5}}{40} - \frac{5 x^{4}}{24} - \frac{x^{3}}{3} + x + 1\right) + C_{1} x^{2} \left(\frac{x^{3}}{6} + \frac{x^{2}}{2} + x + 1\right) + O\left(x^{6}\right)$$