Let k be a positive integer and y be a column vector of functions with length 2k. We have the system of differential equations y' = My, = where Mi,k+iMk+i,i = i for 1
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- Find the general solution of the differential systemVerify that the vector X is a solution of the given homogeneous system. 1 -1 -5t/4 * - (- ² ² 6 )*; x = (-²) 0-²4²4 X' = 16 X; X 1 -1. = (-₁) e For X = (- : 1 -5t/4 1 , one has X'= 16 X= 0000Consider the system of differential equations yi Ул — 292, Yı + 4y2. Rewrite the equations in vector form as j'(t) = Ağ(t). %3D j'(t) = T(t)
- Write the given second order equation as its equivalent system of first order equations. u" + 4.5u' – 7.5u = -7 sin(3t), u(1) = -7.5, u'(1) = -6 Use v to represent the "velocity function", i.e. v = u'(t). Use v and u for the two functions, rather than u(t) and v(t). (The latter confuses webwork. Functions like sin(t) are ok.) u' = %3D Now write the system using matrices: u d + dt V V and the initial value for the vector valued function is: u(1) v(1)= - 5x' + 3x' + x + t into a system of first order equations Convert the differential equation x by letting u₁ = x, U2 = x', Uz = x'' Note: Type u_1, u_2, u_3, and t to write your variables. Ղալ = U2 Uzconvert the given linear differential equations to a first-order linear system. y" + 2ty' + y = cost
- Convert the differential equation u"-3u-2u: = est into a system of first order equations by letting x = u, y = u' x' = y' =Solve the system Ÿ' = AŸ + Ġ for AConsider a disease where immunity is lost. On average, aftern days, a recovered person becomes susceptible again. Adjust the equationsfor ds/dt , di/dt, and dr/dt in the SIR model to account for this.
- Let k be a positive integer and y be a column vector of functions with length 2k. We have the system of differential equations y' = My, where Mi,k+iMk+iii for 1≤i≤k (all other entries of M are 0). What is the solution to this system of differential equations?Find two linearly independent solutions of 2a?y" – xy' + (5x + 1)y= 0, x > 0 of the form Y1 = x" (1+ a1r + a2x? + a3x³+..) Y2 = x" (1+ bịT + bzx² + b3x³+..) where r1 > r2- Enter T1 a1 a2 a3 r2 = %3D b2 b3Convert the differential equation x'''=x''−2x'−4x+tx′′′=x′′-2x′-4x+t into a system of first order equations by letting u1=x,u2=x',u3=x''u1=x,u2=x′,u3=x′′Note: Type u_1, u_2, u_3, and t to write your variables.u1'u1′ = u2'u2′ = u3'u3′ = Submit QuestionQuestion 3