2. Convert the following 2nd order ODEs into a system of 1st order ODEs . - (a) ThP (T-T∞) = 0 kAc (b) x²+xy+ y = 0

Put the following second-order ODEs into the form y" = F(x, y, y') and de- termine a change of variables to reduce the system to a first-order ODE. (a) (b) (c) d²y dr² x dy -2 = da dr² d²y dr² x 12x², 3 (12) ³ = 0, + ²1 (1²) ² = = 0.

Which of the following is a linear system of ODEs? x'=7sin(x)−7y,  y'=7x−2cos(y)x′=7sin(x)-7y,  y′=7x-2cos(y) x'=−2tx+2ty+7t,  y'=−2tx+7t−2y−ln(t)x′=-2tx+2ty+7t,  y′=-2tx+7t-2y-ln(t) x'=−2x+2xy+sin(t),  y'=7xy−2y−cos(t)x′=-2x+2xy+sin(t),  y′=7xy-2y-cos(t) x'=−2x−5y+sin(ty),  y'=−2x−2y−cos(tx)x′=-2x-5y+sin(ty),  y′=-2x-2y-cos(tx) x'=7x+2y+sin(t),  y'=7x−2y−cos(t)

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Let h(n) = 3(1/2)² u(n) – 2(1/3)"-1 u(n) be the unit sample response of a linear shift-invariant system. If the input to this system is a unit step, n 20 x(n) = else find lim,o y(n) where y(n) = h(n) * x(n). -3 -2 O 1 0

Consider the system of linear ODES (3₁ (t)) The system of equation written as y' (t) = Ay(t), where y(t) = 12 (1) 93 (t) (a) Enter the matrix A in the box below. sin (a) 8 dr v₁ (t)= 4 Ae-5t-5 Belt +Ce-t 1/2 (t)= v3 (t). 91 (t)=33 y₁ (t)-9 y2 (t) + 25 y/3 (t). 92 (t)=38 y₁ (t)-14 y2 (t) + 25 y3 (t). 3 (t)=-20 y1 (t)-21 yz (t). f (b) Given that the matrix A has eigenvectors 3 20 60 8 000 with respective eigenvalues -5,4,-1. Enter the solution of 32(t) and 33 (t) in Maple syntax, making sure the answer is consistent with the given form of y₁ (t). a Ω E

1. Turn the following 2nd order ODE: where y' = dy/dx у" - бу' + 5y %3 0 into a dynamical system of 1st order ODES with independent variable t.

Need help with this Linear First Order Mixing Problem. Thank you!

(5) Write the following second order ODE as a system of 2 first-order ODES: 2 0= d²₂ 'I + dt² m a³ da dt (1)

3. Consider the linear ODE: -2 -1 dt dr2 dt 12 find the general solution draw the phase protrait