作业帮常微分方程 欧拉方程 推导
来源：学生作业帮 编辑：拍题作业网作业帮 分类：数学作业 时间：2024/04/30 10:15:29
常微分方程 欧拉方程 推导
常微分方程 欧拉方程 有这样一步
令x=e^t t=lnx
如何推导出d^2y/dx^2和d^3y/dx^3的关于t的二阶三阶导数表达式
常微分方程 欧拉方程 有这样一步
令x=e^t t=lnx
如何推导出d^2y/dx^2和d^3y/dx^3的关于t的二阶三阶导数表达式
dy/dx=(dy/dt)*(dt/dx)=1/e^t*(dy/dt)
d^2y/dx^2={d[1/e^t*(dy/dt)]/dt}*(dt/dx)
=(1/e^t)*(d^2y/dt^2-dy/dt)*(1/e^t)
=(1/e^t)^2*(d^2y/dt^2-dy/dt)
d^3y/dx^3={d[(1/e^t)^2*(d^2y/dt^2-dy/dt)]/dt}*(dt/dx)
=[(1/e^t)^2*(d^3y/dt^3-d^2y/dt^2)-2(1/e^t)^2*(d^2y/dt^2-dy/dt)]*(1/e^t)
=(1/e^t)^3*(d^3y/dt^3-3d^2y/dt^2+2dy/dt)
d^2y/dx^2={d[1/e^t*(dy/dt)]/dt}*(dt/dx)
=(1/e^t)*(d^2y/dt^2-dy/dt)*(1/e^t)
=(1/e^t)^2*(d^2y/dt^2-dy/dt)
d^3y/dx^3={d[(1/e^t)^2*(d^2y/dt^2-dy/dt)]/dt}*(dt/dx)
=[(1/e^t)^2*(d^3y/dt^3-d^2y/dt^2)-2(1/e^t)^2*(d^2y/dt^2-dy/dt)]*(1/e^t)
=(1/e^t)^3*(d^3y/dt^3-3d^2y/dt^2+2dy/dt)