a*********3
发帖数: 660 | 1 多谢！
y"+y-y^3=0. Find the set of initial condition y(0),y'(0)for which y(t)has
nonconstant periodic solutions |
f*****e
发帖数: 2992 | 2 Laplace or 傅里叶变换求根，根是纯虚数。
【在 a*********3 的大作中提到】
: 多谢！
: y"+y-y^3=0. Find the set of initial condition y(0),y'(0)for which y(t)has
: nonconstant periodic solutions
|
f*c
发帖数: 687 | 3 Let q=y', then q'=y^3-y. Work on the y-q plane.
Observe that any level curves of
F(y, q)=0.5 q^2 - 0.25(y^2-1)^2
is a solution to your ODE. Most such curves are regular. The
separatrix are the parabolas
q=(y^2-1)/(\sqrt 2) and q=-(y^2-1)/(\sqrt 2).
You can check that if you start at any point inside the domain
bounded by these two curves, i.e.
-(y^2-1)/(\sqrt 2)< q <(y^2-1)/(\sqrt 2) with -1
you get periodic solutions. Exclude (0, 0) if you don't want constant
solution.
|
R******o
发帖数: 1572 | 4 dont think laplace works
do u know how to transform y^3 ?
【在 f*****e 的大作中提到】
: Laplace or 傅里叶变换求根，根是纯虚数。
|
s*****e
发帖数: 115 | 5 找首次积分，观察相空间结构，基本上是这类问题的标准方法。
看来楼主得加紧复习了！
【在 f*c 的大作中提到】
: Let q=y', then q'=y^3-y. Work on the y-q plane.
: Observe that any level curves of
: F(y, q)=0.5 q^2 - 0.25(y^2-1)^2
: is a solution to your ODE. Most such curves are regular. The
: separatrix are the parabolas
: q=(y^2-1)/(\sqrt 2) and q=-(y^2-1)/(\sqrt 2).
: You can check that if you start at any point inside the domain
: bounded by these two curves, i.e.
: -(y^2-1)/(\sqrt 2)< q <(y^2-1)/(\sqrt 2) with -1: you get periodic solutions. Exclude (0, 0) if you don't want constant
|
f*****e
发帖数: 2992 | 6 确实，这个是非线性的。;-(
【在 R******o 的大作中提到】
: dont think laplace works
: do u know how to transform y^3 ?
|
