求恰当微分方程的解验证此方程是恰当微分方程,并求出方程的解(y^2/(x-y)^2-1/x)dx+(1/y-x^2/(x
来源:学生作业帮 编辑:百度作业网作业帮 分类:数学作业 时间:2024/07/25 13:22:41
求恰当微分方程的解
验证此方程是恰当微分方程,并求出方程的解
(y^2/(x-y)^2-1/x)dx+(1/y-x^2/(x-y)^2)dy=0
其中数字2全为上标.
验证此方程是恰当微分方程,并求出方程的解
(y^2/(x-y)^2-1/x)dx+(1/y-x^2/(x-y)^2)dy=0
其中数字2全为上标.
![求恰当微分方程的解验证此方程是恰当微分方程,并求出方程的解(y^2/(x-y)^2-1/x)dx+(1/y-x^2/(x](/uploads/image/z/17239341-21-1.jpg?t=%E6%B1%82%E6%81%B0%E5%BD%93%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%E7%9A%84%E8%A7%A3%E9%AA%8C%E8%AF%81%E6%AD%A4%E6%96%B9%E7%A8%8B%E6%98%AF%E6%81%B0%E5%BD%93%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8B%2C%E5%B9%B6%E6%B1%82%E5%87%BA%E6%96%B9%E7%A8%8B%E7%9A%84%E8%A7%A3%EF%BC%88y%5E2%2F%28x-y%29%5E2-1%2Fx%EF%BC%89dx%2B%281%2Fy-x%5E2%2F%28x)
解法一:凑微分
(y²/(x-y)²-1/x)dx+(1/y-x²/(x-y)²)dy=0
(y²dx-x²dy)/(x-y)²+d(lny-lnx)=0
(dx/x²-dy/y²)/(1/y-1/x)²+dln(y/x)=0
d(1/y-1/x)/(1/y-1/x)²+din(y/x)=0
d(1/(1/x-1/y))+dln(y/x)=0
d(xy/(y-x)+ln(y/x))=0
两边积分得xy/(x-y)+ln(y/x)=C
解法二:求偏导,以下用δx,δy表示相应的偏导符号
(δy)(y^2/(x-y)^2-1/x)=2y/(x-y)²+2y²/(x-y)³=2xy/(x-y)³
(δx)(1/y-x²/(x-y)²)=-2x/(x-y)²+2x²(x-y)³=2xy/(x-y)³
两个偏导相等,故左边为一个全微分,方程为恰当微分方程.
(y²/(x-y)²-1/x)dx+(1/y-x²/(x-y)²)dy=0
(y²dx-x²dy)/(x-y)²+d(lny-lnx)=0
(dx/x²-dy/y²)/(1/y-1/x)²+dln(y/x)=0
d(1/y-1/x)/(1/y-1/x)²+din(y/x)=0
d(1/(1/x-1/y))+dln(y/x)=0
d(xy/(y-x)+ln(y/x))=0
两边积分得xy/(x-y)+ln(y/x)=C
解法二:求偏导,以下用δx,δy表示相应的偏导符号
(δy)(y^2/(x-y)^2-1/x)=2y/(x-y)²+2y²/(x-y)³=2xy/(x-y)³
(δx)(1/y-x²/(x-y)²)=-2x/(x-y)²+2x²(x-y)³=2xy/(x-y)³
两个偏导相等,故左边为一个全微分,方程为恰当微分方程.
求微分方程(y/x-1)dy/dx=(y/x)^2的解
高数-常微分方程验证下题的函数是否为相应方程的解,并指出是通解还是特解(x-2y)y'= 2x-y 由方程x^2-xy+
验证函数y=(c1+c2*x)e^2x是微分方程y"-4y'+4y=0的通解,并求次微分方程满足初值条件y(0)=1,y
求微分方程的通解:x(y^2-1)dx+y(x^2-1)dy=0
求下列微分方程的解(1)(xy+x^3y)dy-(1+y^2)dx=0 (2)(y^2-6x)y'+2y=0
高数微分方程,已知y=1 y=x y=x^2 是某二阶非齐次线性微分方程的三个解,则该方程的通解为______
微分方程通解问题已知y=1,y=x,y=x^2是某二阶非齐次线性微分方程的三个解,则该方程的通解为?
求下列微分方程的特解:dy/dx=y/2根号x,y|x=4=1
求微分方程x^3*(dy/dx)=x^2*y-1/2*y^3满足初始条件y|(x=1)=1的特解
求微分方程的通解.[1+2e^(x/y)]dx+ 2e^(x/y)*[1-x/y]dy=0.
求微分方程dy\dx=2x-y的通解
求微分方程y''-3y'+2y=2e^x满足y|x=0 =1,dy/dx|x=0 =0的特解