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\(y(0)=0\)\(y'(0)=1\)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\(x'=-xy^2+4x^3y^2\)\(y'=-y+y^3\).

(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\(T\)\(A(t+T)=A(t)\),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\(T\)\(A(t+T)=A(t)\).

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\(\psi(x)\equiv0\).


Last update: July 26, 2020