Exercise

1. Find singular points of the following differential equations and classify.
$(a) \ x^2 y^{\prime\prime} + xy^{\prime} + y = e^{x}$
$(b) \ x(x-1)^{3}y^{\prime\prime} + xy^{\prime} + (x-1)^{2}y = 0$
$(c) \ (2 - x)y^{\prime\prime} + xy^{\prime} + \frac{y}{(x-2)^{2}} = 0$
2. Find a solution of the following differential equation around $x = 0$

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$(a) \ 2xy^{\prime\prime} + 3y^{\prime} - y = 0$
$(b) \ xy^{\prime\prime} + y^{\prime} + xy = 0$
$(c) \ 4x^2 y^{\prime\prime} - 2x(x-2)y^{\prime} - (3x+1)y = 0$
3. Find the solutions of the following differential equations around the designated point.
$(a) \ (1-x^2)y^{\prime\prime} -2xy^{\prime} + n(n+1)y = 0, \ a= 0$
This equation is called Legendre's equation
$(b) \ (1-x^2)y^{\prime\prime} -2xy^{\prime} + 12y = 0, \ a= 1$
$(c) \ x^2 y^{\prime\prime} + xy^{\prime} + (x^2 -\nu^{2}) y = 0, \ a = 0$
This is called Bessel's equation.