2016-2017第一学期常微分方程期末考试

一、解方程x^{(6)}+6x^{(4)}+9x''=0.

二、解方程

x'=\left( \begin{array}{cc}-2&3\\-1&2\end{array}\right)x+\left(\begin{array}{c}t\\1\end{array} \right).

三、解方程(t^2+1)x''+4tx'+2x=0,已知有特解x=\dfrac{1}{t^2+1}.

四、解方程x''+\dfrac{2}{1-x}(x')^2=0.

五、设y=\varphi(x)是方程y''+ay'+by=0满足初值条件y(0)=0y'(0)=1的解,证明:\displaystyle y=\int_{0}^{x}\varphi(x-t)f(t)\mathrm{d}t是方程y''+ay'+by=f(x)的解.>

六、已知方程x'=-xy^2+4x^3y^2y'=-y+y^3.

(1)利用V(x,y)=x^2+y^2判定方程零解的稳定性.

(2)判定解x\equiv \dfrac{1}{2}y\equiv1的稳定性.

七、设\varPhi是方程x'=A(t)x的基解矩阵.证明:

(1)若A(t)有周期TA(t+T)=A(t),则存在常数矩阵B使得\varPhi(t+T)=\varPhi(t)B.

(2)若存在常数矩阵B使得\varPhi(t+T)=\varPhi(t)B,则A(t)有周期TA(t+T)=A(t).

八、设\varphi(x),\psi(x)是方程x''+p(t)x'+q(t)=0的解,其中\varphi(x)满足\varphi(a)=\varphi(b)=0,且在(a,b)内恒不为0.证明:

(1)\psi(x)[a,b]有零点.

(2)若\psi(x)[a,b]有3个零点,则\psi(x)\equiv0.