I would like to know the correct answers and how you came up with it. (1) Rewrit
ID: 1945155 • Letter: I
Question
I would like to know the correct answers and how you came up with it.
(1) Rewrite the second order differential equation d^2(x)/dt - 1/2(dx/dt)^2+sin(x)=0 as a first order system.
(a). du/dt=v,dv/dt=-sin(u)+v^2/2
(b). du/dt=v,dv/dt=-sin(v)-v^2/2
(c). du/dt=tsin(u)+v^2/2
(d). du/dt=tsin(u)+t
(2). Rewrite the second order differential equation d^2(x)+dx/dt+mx=0, where x(0)=x0, dx/dt=y0.
(a). du/dt=v,dv/dt=-v+mu,where u(0)=xo,v(0)=y0
(b).du/dt=v,dv/dt=-v-mu,where u(0)=xo,v(0)=y0
(c). du/dt=v,dv/dt=-mu+v,where u(0)=x0,v(0)=y0
Explanation / Answer
I would like to know the correct answers and how you came up with it.
(1) Rewrite the second order differential equation d^2(x)/dt - 1/2(dx/dt)^2+sin(x)=0 as a first order system.
METHOD ..
[' DENOTES FORST DERIVATIVE .......'' DENOTES SECOND DERIVATIVE]
1.PUT
U=X.........................................1
DIFFERENTIATING WRT T
X'=U'
PUT
V=U'=X'...............................................2
DIFFERENTIATING AGAIN WRT T
V'=X''..........................................3
PUTTING 1,2,3 IN THE GIVEN D.E. WE GET
V'-0.5V^2+SIN(U)=0...........
V'=DV/DT = 0.5V^2-SIN(U)........................4
THIS AND THE EQN.2..VIZ
V=U' = DU/DT ..................................2
FORM THE SYSTEM OF EQNS.
HENCE (a). du/dt=v,dv/dt=-sin(u)+v^2/2...IS THE RIGHT CHOICE
(a). du/dt=v,dv/dt=-sin(u)+v^2/2................CORRECT
(b). du/dt=v,dv/dt=-sin(v)-v^2/2
(c). du/dt=tsin(u)+v^2/2
(d). du/dt=tsin(u)+t