Question

高数？。。。

第十题怎么整？？。。。？

Answer 1

x=e^t*sint，y=e^t*cost
dx/dt=e^t*(sint+cost)，dy/dt=e^t*(cost-sint)
dy/dx=(cost-sinx)/(sint+cost)

d(dy/dx)/dt=-[(sint+cost)^2+(cost-sint)^2]/(sint+cost)^2
d²y/dx²=d(dy/dx)/dx
=[d(dy/dx)/dt]/(dx/dt)
=-[(sint+cost)^2+(cost-sint)^2]/[e^t*(sint+cost)^3]
左边=d²y/dx²*(x+y)^2
=-[(sint+cost)^2+(cost-sint)^2]/[e^t*(sint+cost)^3]*(sint+cost)^2*e^(2t)
=-[(sint+cost)^2+(cost-sint)^2]/(sint+cost)*e^t
=-2(e^t)/(sint+cost)
右边=2(x*dy/dx-y)
=2[e^t*sint*(cost-sinx)/(sint+cost)-e^t*cost]
=2e^t*[sint*(cost-sinx)/(sint+cost)-cost]
=2e^t*[-1/(sint+cost)]
=-2(e^t)/(sint+cost)

所以左边=右边，证毕