Determine a lower bound for the radius $of$ convergence $of$ series

solutions about the given $x_0$ for each $of$ the differential equations

given below:

$(a)y^{\text{'}\text{'}}-4y=0,x_0=0$

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$(b)y^{\text{'}\text{'}}+4y^{\text{'}}=0,x_0=0$

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$(c)y^{\text{'}\text{'}}-4x{y}^{\text{'}}-11y=0,x_0=0$

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$(d)y^{\text{'}\text{'}}-4x{y}^{\text{'}}-11y=0,x_0=1$

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$(e)y^{\text{'}\text{'}}-4k^2{x}^{2}y=0,x_0=0,k$ constant

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$(f)(1-x)y^{\text{'}\text{'}}+4y=0,x_0=0$

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$(g)y^{\text{'}\text{'}}+4x{y}^{\text{'}}+11y=0,x_0=0$

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$(h)x{y}^{\text{'}\text{'}}+4y^{\text{'}}+11xy=0,x_0=1$

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$(i)(3-x^2)y^{\text{'}\text{'}}-4x{y}^{\text{'}}-11y=0,x_0=0$

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$(j)2y^{\text{'}\text{'}}+4x{y}^{\text{'}}+11y=0,x_0=0$

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$(k)4y^{\text{'}\text{'}}+(x+1)y^{\text{'}}+11y=0,x_0=2$

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