First Order ODEs (cont). Modeling with First Order ODEs September 11 15, 2017
Bernoulli s ODEs Yuliya Gorb Definition A first order ODE is called a Bernoulli s equation iff it is written in the form y +p(t)y = q(t)y κ, for κ 0,1, (1) with continuous functions p(t), q(t) on some interval (a, b) Examples: y +y 2 = 0 Bernoulli s eq. y 4y = 2e x y Bernoulli s eq. y +p(t)y = q(t)y not a Bernoulli s eq. but linear y +t = y 2 not a Bernoulli s eq.
Bernoulli s ODEs (cont.) Nice feature of a Bernoulli s ODE is that using substitution z = y 1 κ one can reduce it to a linear one. Indeed, multiply both sides of (1) by y κ : and evaluate y κ y +p(t)y 1 κ = q(t), (2) z = (1 κ)y κ y y κ y = z 1 κ Now substitute it into (2): that leads to z +p(t)z = q(t), 1 κ z +(1 κ)p(t)z = (1 κ)q(t), which is linear
Bernoulli s ODEs (cont.) Example: { y Solve +t 1 y = ty 2 y(1) = 1 Observe that this is Bernoulli s ODE with κ = 2, then multiply both sides of the ODE by y 2 y 2 y +t 1 y 1 = t. Now substitute z = y 1 then z = y 2 y back into the ODE: z + 1 z = t, and t obtain: z 1 z = t, and this is a linear ODE (3) t The integrating factor then is µ(t) = e dt t = 1. Multiply both sides of (3) by this t integrating factor and have d [ z ] = 1 dt t z Integrate both sides: = t +C and substitute back for y to obtain the general t 1 solution of the original ODE: yt = t +C y = 1 t 2 +Ct. 1 Use the given IC: 1 = to have C = 2, and obtain the solution to the IVP: 1+C y = 1 t 2 +2t
Modeling with First Order ODEs Modeling with First Order ODEs you must: 1 Identify all the given parameters of the problem and determine independent and dependent variables 2 Choose a governing principle that describes the problem at hand 3 Check the consistency of units 4 Derive a first order ODE and identify the initial condition 5 Solve the obtained IVP and answer a specific question of the problem
Falling Object Problem Revisited Example: A ball is thrown upward with initial velocity 20 m/sec from the roof of a building 30 m high. Neglect air resistance. 1 Find the maximum height above the ground that the ball reaches. 2 Assuming that the ball misses the building on the way down, find the time that it hits the ground. t time, [s] (independent variable) v, V velocity, v = v(t), [m/s] (dependent variable) h, H height, h = h(t), [m] (another dependent variable) m mass, [kg/s] g acceleration due to gravity, g = 9.8 [m/s 2 ] Here, we take a different approach from one in class. Namely, now I assume that the positive direction is upward with the origin on the ground level 1 First, using the Newton s 2nd Law set a model for the ball going upward: m dv = mg dt v(0) = 20 whose solution is v(t) = 9.8t +20 m/sec
Falling Object Problem Revisited (cont.) Note that mass canceled out because of which it was not given Now find a moment of time T when this velocity is zero: v(t) = 9.8T +20 = 0 T 2.04 sec And now set the IVP for the height of the current position of the ball above the roof: dh = v(t) = 9.8t +20 dt h(0) = 30 whose solution is h(t) = 4.9t 2 +20t +30 m So, we need to compute the height of the ball above the roof at time T: h(t) 20.4 m taking int account the building s height, the maximal height of the ball above the ground is 50.4 m 2 Need to set a new model for velocity V = V(t) when the ball falls m dv = mg dt V(0) = 0 whose solution is V(t) = 9.8t m/sec
Falling Object Problem Revisited (cont.) Then the height H = H(t) of the ball when it falls satisfies: dh = V(t) = 9.8t dt H(0) = 50.4 whose solution is H(t) = 4.9t 2 +50.4 m And now we need to find time T when the height is 0: H(T ) = 4.9T 2 +50.4 = 0 m T 3.2 sec taking int account the time the ball was going up we find the total travel time as 5.24 sec