2016-2017第一学期常微分方程期中考试

一、解下列微分方程

(1)\(x(y-x)y'=y^2\).

(2)\(y'=-2xy+2x\).

(3)\((3x^2y+2xy+y^3)\mathrm{d}x+(x^2+y^2)\mathrm{d}y=0\).

(4)\(y'(x-\ln y')=1\).

二、已知\(f(x,y)=x\sqrt{|y|}\),证明:

(1)\(f(x,y)\)在区域\(\{-1<x<1,-\infty <y<+ \infty\}\)内对\(y\)不满足局部Lipschitz条件;

(2)\(f(x,y)\)在区域\(\{-1<x<1,0<y<+\infty\}\)内对\(y\)满足局部Lipschitz条件,但不满足Lipschitz条件.

三、设方程\(\displaystyle M(x,y)\mathrm{d}x+N(x,y)\mathrm{d}y=0 \qquad (1)\)中的\(M(x,y),N(x,y)\)连续可微且满足关系\(\displaystyle \dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}=Nf(x)-Mg(y)\), 其中\(f(x),g(y)\)分别为\(x,y\)的连续函数,证明方程(1)有积分因子\(\displaystyle \mu =\mathrm{exp}\bigg(\int f(x)\mathrm{d}x+\int g(y)\mathrm{d}y\bigg)\).

四、证明方程\(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{\cos y}{e^x+y^2}\)的每一个解的最大存在区间都是\((-\infty , +\infty)\).

五、设\(f_1(x,y),f_2(x,y)\)在区域\(R:|x|\leqslant a,|y|\leqslant b\)连续可微,且满足\(\displaystyle f_1(x,y)<f_2(x,y),\forall (x,y) \in R.\)证明:若\(\varphi_1(x),\varphi_2(x)\)是方程\(y'=f_1(x,y),y'=f_2(x,y)\)过初值\((0,0)\)的解,则当\(0< x \leqslant h\)时有\(\varphi_1(x)<\varphi_2(x)\).其中\(h=\min\{a,\dfrac{b}{M_1},\dfrac{b}{M_2}\}\)\(M_i=\mathop{\max}\limits_{(x,y)\in R}|f_i(x,y)|(i=1,2)\).


Last update: July 26, 2020