In[]:=
Remove["Global`*"]
Problem 1
In[]:=
DE:=(1-x)y''[x]+xy'[x]-y[x]==0​​BC:={y[0]==-3,y'[0]==2}​​ϕ[xd_]:=(y[x]/.DSolve[{DE,BC},y[x],x][[1]][[1]])/.xxd
In[]:=
ϕ[x]
Out[]=
-3
x

+5x
In[]:=
Series[ϕ[x],{x,0,4}]
Out[]=
-3+2x-
3
2
x
2
-
3
x
2
-
4
x
8
+
5
O[x]
Problem 2
In[]:=
DE:=y''[x]+xy[x]==0​​BC:={y[0]==1,y'[0]==0}​​ysol[xd_]:=(y[x]/.DSolve[{DE,BC},y[x],x][[1]])/.xxd
In[]:=
ysol[x]
Out[]=
1
2
2/3
3
AiryAi
1/3
(-1)
xGamma
2
3
+
1/6
3
AiryBi
1/3
(-1)
xGamma
2
3

In[]:=
Series[ysol[x],{x,0,6}]
Out[]=
1-
3
x
6
+
6
x
180
+
7
O[x]
Problem 3
In[]:=
DE:=2x^2y''[x]+3xy'[x]-y[x]==0​​ysol[xd_]:=(y[x]/.DSolve[DE,y[x],x][[1]])/.xxd
In[]:=
ysol[x]
Out[]=
x
C[1]+
C[2]
x
In[]:=
Wronskian[ysol[x]/.{{C[1]1,C[2]0},{C[1]0,C[2]1}},x]
Out[]=
-
3
2
3/2
x
Problem 4
In[]:=
DE:=x^2y''[x]+5xy'[x]+4y[x]==0​​ysol[xd_]:=(y[x]/.DSolve[DE,y[x],x][[1]])/.xxd
In[]:=
ysol[x]
Out[]=
C[1]
2
x
+
2C[2]Log[x]
2
x