1)微分方程y'=xy的通解?2)y'=2y 满足初始条件y'(0)=2 3)lim n趋向于无穷 (n^2/1-n)sin(1/n)=?
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![1)微分方程y'=xy的通解?2)y'=2y 满足初始条件y'(0)=2 3)lim n趋向于无穷 (n^2/1-n)sin(1/n)=?](/uploads/image/z/10230352-16-2.jpg?t=1%EF%BC%89%E5%BE%AE%E5%88%86%E6%96%B9%E7%A8%8By%27%3Dxy%E7%9A%84%E9%80%9A%E8%A7%A3%3F2%EF%BC%89y%27%3D2y+%E6%BB%A1%E8%B6%B3%E5%88%9D%E5%A7%8B%E6%9D%A1%E4%BB%B6y%27%280%29%3D2+3%EF%BC%89lim+n%E8%B6%8B%E5%90%91%E4%BA%8E%E6%97%A0%E7%A9%B7+%28n%5E2%2F1-n%29sin%281%2Fn%29%3D%3F)
1)微分方程y'=xy的通解?2)y'=2y 满足初始条件y'(0)=2 3)lim n趋向于无穷 (n^2/1-n)sin(1/n)=?
1)微分方程y'=xy的通解?
2)y'=2y 满足初始条件y'(0)=2
3)lim n趋向于无穷 (n^2/1-n)sin(1/n)=?
1)微分方程y'=xy的通解?2)y'=2y 满足初始条件y'(0)=2 3)lim n趋向于无穷 (n^2/1-n)sin(1/n)=?
1)∵y'=xy ==>dy/dx=xy
==>dy/y=xdx
==>ln│y│=x²/2+ln│C│ (C是积分常数)
==>y=Ce^(x²/2)
∴原方程的通解是y=Ce^(x²/2) (C是积分常数);
2)∵y'=2y ==>dy/dx=2y
==>dy/y=2dx
==>ln│y│=2x+ln│C│ (C是积分常数)
==>y=Ce^(2x)
∴原方程的通解是y=Ce^(2x) (C是积分常数)
==>y'=2Ce^(2x)
∵y'(0)=2 ==>2C=2
==>C=1
∴y'=2y 满足初始条件y'(0)=2 的特解是y=e^(2x);
3)原式=lim(n->∞){[n/(1-n)]*[sin(1/n)/(1/n)]}
={lim(n->∞)[n/(1-n)]}*{lim(n->∞)[sin(1/n)/(1/n)]}
={lim(n->∞)[1/(1/n-1)]}*{lim(n->∞)[sin(1/n)/(1/n)]}
=[1/(0-1)]*1 (应用重要极限lim(x->0)(sinx/x)=1)
=-1.
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