Ordinary Differential Equations
Differential equations of the form y=xy’+g(y’) with general solution yA=xK+g(K), K Î R. Determine the
singular solutions yS which are not included in the general solutions. Plot the family of curves yA and the
solution yS.
Example 2.6.1
y = xy ’ + Hy ’L2
Ÿ Solution
In[1]:=
Out[1]=
DSolve@y@xD Š x * y ’@xD + Hy ’@xDL ^ 2, y@xD, xD
2
99y@xD ® x C@1D + C@1D ==
Ÿ Plot
In[2]:=
Plot@8Table@8x * c1 + c1 ^ 2<, 8c1, -3, 3, 0.5<D, -x ^ 2  4<,
8x, -5, 5<, PlotStyle ® Thick, PlotRange ® 3, Background ® WhiteD
3
2
1
Out[2]=
-4
2
-2
-1
-2
-3
4
2 beispiel_2_6.nb
In[3]:=
Manipulate@Plot@8x * c1 + c1 ^ 2, -x ^ 2  4<, 8x, -8, 8<,
PlotStyle ® Thick, PlotRange ® 8-5, 5<, Background ® WhiteD, 88c1, 0<, -3, 3<D
c1
4
2
Out[3]=
5
-5
-2
-4
Example 2.6.2
1
y = xy ’ + €€€€€€€
y’
Ÿ Solution
In[4]:=
Out[4]=
DSolve@y@xD Š x * y ’@xD + 1  y ’@xD, y@xD, xD  FullSimplify
1
::y@xD ® €€€
2
4x
1 ãC@1D
2x
i
y
C@1D
j
j
z
+ €€€€€€€€€€€€€€€€€€€€€€€€ z
j-1 + ã
z>, :y@xD ® - €€€ - €€€€€€€€€€€€ - €€€€€€€€€€€€€€€€€€€€€ >>
C@1D
2
2
-1 + ã
1 + ãC@1D
k
{
beispiel_2_6.nb 3
Ÿ Plot
In[5]:=
ManipulateB
1
PlotB: €€€
2
4x
1 ãC@1D
2x
i
y
C@1D
j
j
z
+ €€€€€€€€€€€€€€€€€€€€€€€€ z
j-1 + ã
z, - €€€ - €€€€€€€€€€€€ - €€€€€€€€€€€€€€€€€€€€€ , 2 * Sqrt@xD, -2 * Sqrt@xD>, 8x, -5, 7<,
C@1D
2
2
-1 + ã
1 + ãC@1D
k
{
PlotRange ® 7, PlotStyle ® 8Thick, Thick, 8Thick, Gray<, 8Thick, Gray<<F, 88C@1D, 1<, -3, 3<F
C@1D
6
4
2
Out[5]=
-4
2
-2
-2
-4
-6
4
6