How to Solve Second Order Differential Equations Involving Sines and Cosines?

  • Thread starter Thread starter tuananh3ap
  • Start date Start date
  • Tags Tags
    Second order
AI Thread Summary
To solve the second-order differential equation x'' = (k/m)x, recognize that it typically yields solutions involving sine and cosine functions. The general solution is a linear combination of these functions, represented as x = A sin(kt) + B cos(kt). An alternative method involves using complex numbers by assuming a trial solution of the form x = e^(λt), which leads to an equation for λ whose roots provide the independent solutions. Understanding complex numbers and Euler's formula is crucial for this approach. Engaging with these concepts will facilitate a clearer path to solving the equation.
tuananh3ap
Messages
18
Reaction score
0
hello every body .I have some problem :
i don't know slove the equation
x"=k/m*x (1)
i don't understand result of (1) is x=Asinkt+Bcoskt
 
Physics news on Phys.org
There are two independent solutions, because it is a second order equation. They are sine and cosine. The general solution is then a linear combination of both, as you can check by just plugging it into the differential equation.
 
i know this thing. But i don't know the way to slove this equation
who can help me. thank you very much
 
Well, one way is to recognise the form: if it involves just x'' and x, then it is usually something with sines and cosines, because those are the two functions you know which are (almost) their own derivatives.

If you know about complex numbers, there is a more elegant way (which also works for more general equations). That is to plug in a trial solution x = e^{\lambda t}. Then the differential equation will give you an equation for \lambda, whose roots provide the n independent solutions (for an n-th order differential equation, in this case, n = 2).
I can work out the example for you, but you better tell me whether you know about complex numbers and stuff like
e^{i\theta} = \cos\theta + i \sin\theta
before I do a lot of work for nothing :smile:
 
TL;DR Summary: I came across this question from a Sri Lankan A-level textbook. Question - An ice cube with a length of 10 cm is immersed in water at 0 °C. An observer observes the ice cube from the water, and it seems to be 7.75 cm long. If the refractive index of water is 4/3, find the height of the ice cube immersed in the water. I could not understand how the apparent height of the ice cube in the water depends on the height of the ice cube immersed in the water. Does anyone have an...
Thread 'Variable mass system : water sprayed into a moving container'
Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
Back
Top