已知x=et 和y=sint 求d2y/dx2
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x=sint
dx=costdt
dx²=-sintdt²
(1-x2)*d2y/dx2-x*dy/dx-y=0化成
(1-sin²t)*d²y/(-sint)dt²-x*dy/sintdt-y=0
(1-sin²t)/sint*d²y/dt²-x/sint *dy/dt-y=0
dy/dx=cost/e^t
d^2y/dx^2=d(cost/e^t)/dx=d[cost*e^(-t)]/(e^tdt)=[e^(-t)dcost+costde^(-t)]/(e^tdt)
=[-sinte^(-t)dt-coste^(-t)dt]/(e^tdt)=-(sint+cost)e^(-t)/e^t
=-(sint+cost)e^(-2t)
郯孟苏乐:[答案] dx=e^tdt,dy=costdtdy/dx=cost/e^td^2y/dx^2=d(cost/e^t)/dx=d[cost*e^(-t)]/(e^tdt)=[e^(-t)dcost+costde^(-t)]/(e^tdt)=[-sinte^(-t)dt-coste^(-t)dt]/(e^tdt)=-(sint+cost)e^(-t)/e^t=-(sint+cost)e^(-2t)
越西县19380643408: x=(e^t)sint y=(e^t)cost 求d^2y/dx^2 - ?
郯孟苏乐: 解:dx/dt=(e^t)sint+(e^t)cost=(e^t)(sint+cost) dy/dt=(e^t)cost-(e^t)sint=(e^t)(cost-sint) dy/dx=(dy/dt)/(dx/dt)=(cost-sint)/(sint+cost) d^2y/dx^2=d(dy/dx)/dx=[d(dy/dx)/dt]/[dx/dt]={[-(sint+cost)^2-(cost-sint)^2]/(sint+cost)^2}/[(e^t)(sint+cost)]=-2[e^(-t)]/(sint+cost)^3
越西县19380643408: 已知曲线的参数方程 X=etsint y=etcost (t为参数)(et表示e的t次方)求y对x的导数 - ?
郯孟苏乐:[答案] dx/dt=etsint+etcost dy/dt=etcost-etdint dy/dx=dy/dt/(dx/dt) =(etcost-etdint)/(etsint+etcost) =(cost-sint)/(sint+cost)
越西县19380643408: 求由参数方程x=etsinty=etcost所确定的曲线在t=0所对应的点处的切线方程 - ?
郯孟苏乐: ∵ dy dx = dy/dt dx/dt = et(cost?sint) et(sint+cost) = cost?sint sint+cost 又当t=0时,x=0,y=1, ∴ dy dx |t=0=1, 所求切线方程为x-y+1=0
越西县19380643408: 如果x=cost y=sint 那么求d²y/dx²=(d/dt )(dy/dx) (dt/dx) 中d/dt部分要代入什么 - ?
郯孟苏乐:[答案] d/dt表示对t求导数,具体表达式在其后面,在这里是dy/dx
越西县19380643408: x=(e^t)sint y=(e^t)cost 求d^2y/dx^2 - ?
郯孟苏乐:[答案] dx/dt=(e^t)sint+(e^t)cost=(e^t)(sint+cost) dy/dt=(e^t)cost-(e^t)sint=(e^t)(cost-sint) dy/dx=(dy/dt)/(dx/dt)=(cost-sint)/(sint+cost) d^2y/dx^2 =d(dy/dx)/dx =[d(dy/dx)/dt]/[dx/dt] ={[-(sint+cost)^2-(cost-sint)^2]/(sint+cost)^2}/[(e^t)(sint+cost)] =-2[e^(-t)]/(sint+cost)^3
越西县19380643408: x=(e^t)sint y=(e^t)cost 求d^2 y/d(x^2)最好有过程,谢谢 - ?
郯孟苏乐: 这个是参数方程求导啊 x't=(e^t)(sint+cost) y't=(e^t)(cost-sint) x''t=(e^t)(sint+cost+cost-sint)=2(e^t)cost y''t=(e^t)(cost-sint-sint+cost)=-(e^t)sint dy/dx=(cost-sint)/(sint+cost) d^2 y/d(x^2)=d(dy/dx)/dx=(y''x'-y'x'')/(x')^2 自己代入吧
越西县19380643408: x=(e^t)sint y=(e^t)cost 求d^2 y/d(x^2)最好有过程, - ?
郯孟苏乐:[答案] 这个是参数方程求导啊 x't=(e^t)(sint+cost) y't=(e^t)(cost-sint) x''t=(e^t)(sint+cost+cost-sint)=2(e^t)cost y''t=(e^t)(cost-sint-sint+cost)=-(e^t)sint dy/dx=(cost-sint)/(sint+cost) d^2 y/d(x^2) =d(dy/dx)/dx =(y''x'-y'x'')/(x')^2 自己代入吧
越西县19380643408: 设(X=TCOST,Y=TSINT,求DY/DX - ?
郯孟苏乐:[答案] 先求dx=(cost-tsint)dt,dy=(sint+tcost)dt 然后dy/dx=(sint+tcost)/(cost-tsint) 根据x=tcost ; y=tsint; y/x=tant 所以dy/dx=[y+yarctan(y/x)]/[x-yarctan(y/x)]
越西县19380643408: 设y=yx, x=sint ,y=arctant,求dy/dx - ?
郯孟苏乐: x=sint dx/dt = costy= arctant dy/dt = 1/(1+t^2)dy/dx = dy/dt . (dt/dx)= 1/[(cost)(1+t^2)]