Krdytkyu

Can we solve analytically (find a closed form solution) the second order ODE

$$x^{alpha}y^{primeprime}=y,quad x>0$$
where $alphain,]0,1[$.
Consider the conditions
$$y(0)=1,quad y^{prime}(0)=1$$
This equation appears in a fractional model that describes viscoelasticity properties in certain materials.

Mathematica gives the answer in terms of the modified Bessel function of the first kind $I_{n}$
http://mathworld.wolfram.com/ModifiedBesselFunctionoftheFirstKind.html

Actually this is the solution of $$x^{1-alpha}y^{primeprime}=y$$

Too long for a comment.

As you wrote in you answer, it seems that you obtained solutions in terms of Bessel $I$ functions (the expressions you gave could be simplified quite a lot). As I wrote in comments, I suppose that they would become even simpler when using the given boundary conditions.

Playing with the case where $alpha=frac mn$ ($m$ and $n$ being integers) and using the boundary conditions, it seems that they also can express in terms of hypergeometric functions where patterns seem to appear (I let you finding them).
$$left(
begin{array}{ccc}
m & n & y(x) \
1 & 2 & , _0F_1left(;frac{1}{3};frac{4
x^{3/2}}{9}right)+x , _0F_1left(;frac{5}{3};frac{4
x^{3/2}}{9}right)\
1 & 3 & , _0F_1left(;frac{2}{5};frac{9
x^{5/3}}{25}right)+x , _0F_1left(;frac{8}{5};frac{9
x^{5/3}}{25}right)\
1 & 4 & , _0F_1left(;frac{3}{7};frac{16
x^{7/4}}{49}right)+x , _0F_1left(;frac{11}{7};frac{16
x^{7/4}}{49}right)\
1 & 5 & , _0F_1left(;frac{4}{9};frac{25
x^{9/5}}{81}right)+x , _0F_1left(;frac{14}{9};frac{25
x^{9/5}}{81}right)\
1 & 6 & , _0F_1left(;frac{5}{11};frac{36
x^{11/6}}{121}right)+x , _0F_1left(;frac{17}{11};frac{36
x^{11/6}}{121}right)\
1 & 7 & , _0F_1left(;frac{6}{13};frac{49
x^{13/7}}{169}right)+x , _0F_1left(;frac{20}{13};frac{49
x^{13/7}}{169}right) \
2 & 3 & , _0F_1left(;frac{1}{4};frac{9
x^{4/3}}{16}right)+x , _0F_1left(;frac{7}{4};frac{9
x^{4/3}}{16}right)\
2 & 5 & , _0F_1left(;frac{3}{8};frac{25
x^{8/5}}{64}right)+x , _0F_1left(;frac{13}{8};frac{25
x^{8/5}}{64}right)\
2 & 7 & , _0F_1left(;frac{5}{12};frac{49
x^{12/7}}{144}right)+x , _0F_1left(;frac{19}{12};frac{49
x^{12/7}}{144}right)\
3 & 4 & , _0F_1left(;frac{1}{5};frac{16
x^{5/4}}{25}right)+x , _0F_1left(;frac{9}{5};frac{16
x^{5/4}}{25}right)\
3 & 5 & , _0F_1left(;frac{2}{7};frac{25
x^{7/5}}{49}right)+x , _0F_1left(;frac{12}{7};frac{25
x^{7/5}}{49}right)\
3 & 7 & , _0F_1left(;frac{4}{11};frac{49
x^{11/7}}{121}right)+x , _0F_1left(;frac{18}{11};frac{49
x^{11/7}}{121}right)\
3 & 8 & , _0F_1left(;frac{5}{13};frac{64
x^{13/8}}{169}right)+x , _0F_1left(;frac{21}{13};frac{64
x^{13/8}}{169}right)\
4 & 5 & , _0F_1left(;frac{1}{6};frac{25
x^{6/5}}{36}right)+x , _0F_1left(;frac{11}{6};frac{25
x^{6/5}}{36}right)\
4 & 7 & , _0F_1left(;frac{3}{10};frac{49
x^{10/7}}{100}right)+x , _0F_1left(;frac{17}{10};frac{49
x^{10/7}}{100}right)\
4 & 9 & , _0F_1left(;frac{5}{14};frac{81
x^{14/9}}{196}right)+x , _0F_1left(;frac{23}{14};frac{81
x^{14/9}}{196}right)
end{array}
right) $$