1.设dxdt=p,则d²xdt²=pdpdx∵d²xdt²=-w²x==>pdpdx=-w²x==>p²=-w²x²+C1² (C1是积分常数)==>p=±√(C1²-w²x²)=。微积分例子？更多详情请大家跟着小编一起来看看吧！
微积分例子(1)
1.设dxdt=p,则d²xdt²=pdpdx
∵d²xdt²=-w²x
==>pdpdx=-w²x
==>p²=-w²x²+C1² (C1是积分常数)
==>p=±√(C1²-w²x²)
==>dxdt=±√(C1²-w²x²)
==>dx√(C1²-w²x²)=±dt
==>(1w)arcsin(wxC1)=±t+C2 (C2是积分常数)
==>x=(C1w)sin[w(C2±t)]
∴原方程的通解是x=(C1w)sin[w(C2±t)] (C1,C2是积分常数)
2.设x=r*tant,则dx=r*sec²tdt,sint=x√(x²+r²)
故∫dx√(x²+r²)=∫r*sec²tdt(r*sect)
=∫ costdtcos²t
=∫ d(sint)(1-sin²t)
=(12)∫ [1(1+sint)+1(1-sint)]d(sint)
=(12)ln│(1+sint)(1-sint)│+C1 (C1是积分常数)
=ln│x+√(x²+r²)│-ln│r│+C1
=ln│x+√(x²+r²)│+C (C=C1-ln│r│.∵C1是积分常数,∴C也是积分常数).
