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Consider the system of differential equations Find the general solution to this system, given that the general solution to the corresponding homo- geneous system is xh(t) = C₁e²t x₁ (t) = 3x₁ (t) + 2x₂(t) x₂(t) = x1(t) + x₂(t) + e²t x₂ (1)= ae²¹ Given the system of differential equations x₁' (1)=3x₁ (1)+2x₂ (1) x₂ (1) = -x₁ (1) + x₂ (1) e²¹ and the homogeneous solution is sint-cost x₁ (1)=C₁e²¹ cost The objective is to find the general solution to the system. (sint-cost) + C₂e²t (- ^) ( +C₂e²¹ Take derivative. x,' (1)-(20₁) e²¹ Plug the values into the given system. a₁ (3) -(39) + (0) 29₂ Find the particular solution. By using the method of undetermined coefficients, the particular solution is of the form = (a ₁)e²+ 21 3a,e² +2a₂e² -a₁e²1 + a₂e²¹ + e²² (sin! The particular solution is xp The general solution is x(1) = x₁ (1) + x₂ (1) =G₁e² Equating the coefficients. 2a₁ = 3a₁ +2a₂ a₁ +2a₂ = 0 2a₂ = a₁ + a₂ +1 a₁ + a₂ = 1 Solving both equations gives a₁ = 2 and a₂ = -1. - cost - sin sin t -cost-sint sint sint-cost COS/ sint) 2 =(²₁) ₁². e²1 2 J + C₂e³² ( - Sini) + (²1) ² -cost-sint sint elp with this one ind the unique solution to the inhomogeneous system above which satisfies x(0) = =

Consider the system of differential equations Find the general solution to this system, given that the general solution to the corresponding homo- geneous system is xh(t) = C₁e²t x₁ (t) = 3x₁ (t) + 2x₂(t) x₂(t) = x1(t) + x₂(t) + e²t x₂ (1)= ae²¹ Given the system of differential equations x₁' (1)=3x₁ (1)+2x₂ (1) x₂ (1) = -x₁ (1) + x₂ (1) e²¹ and the homogeneous solution is sint-cost x₁ (1)=C₁e²¹ cost The objective is to find the general solution to the system. (sint-cost) + C₂e²t (- ^) ( +C₂e²¹ Take derivative. x,' (1)-(20₁) e²¹ Plug the values into the given system. a₁ (3) -(39) + (0) 29₂ Find the particular solution. By using the method of undetermined coefficients, the particular solution is of the form = (a ₁)e²+ 21 3a,e² +2a₂e² -a₁e²1 + a₂e²¹ + e²² (sin! The particular solution is xp The general solution is x(1) = x₁ (1) + x₂ (1) =G₁e² Equating the coefficients. 2a₁ = 3a₁ +2a₂ a₁ +2a₂ = 0 2a₂ = a₁ + a₂ +1 a₁ + a₂ = 1 Solving both equations gives a₁ = 2 and a₂ = -1. - cost - sin sin t -cost-sint sint sint-cost COS/ sint) 2 =(²₁) ₁². e²1 2 J + C₂e³² ( - Sini) + (²1) ² -cost-sint sint elp with this one ind the unique solution to the inhomogeneous system above which satisfies x(0) = =

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.CR: Chapter 11 Review
Problem 12CR
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Find the unique solution to the inhomogeneous system above which satisfies x(0) = matrix ....

 

Consider the system of differential equations Find the general solution to this system, given that the general solution to the corresponding homo- geneous system is x₁ (t) = 3x₁ (t) + 2x₂(t) x₂ (t) = x1(t) + x₂(t) + e²t xh(t) = C₁e²t (sin x₂ (1)= ae²¹ = (a ₁) ²² Given the system of differential equations x' (1)=3x₁ (1)+2x₂ (1) x₂² (1) = -x₂₁ (1) + x₂ (1) e²¹ and the homogeneous solution is sint-cost -cost-sint x₂ (1)=C₁e²¹ +C₂e²¹ cost sint The objective is to find the general solution to the system. Find the particular solution. By using the method of undetermined coefficients, the particular solution is of the form Take derivative. x,' (1)=(201₁) e²¹ Plug the values into the given system. 3 2 a₁ (29)² =(³, 1)(a)~²¹ +(2₂) -1 3a,e² +2a₂e² -a₁e²² + a₂e²¹ +e²₁ → t - cost cos t Equating the coefficients. 2a₁ = 3a₁ +2a₂ 2a₂ = -a₁ + a₂ +1 a₁ + a₂ = 1 Solving both equations gives a, = 2 and a₂ =-1. st) + C₂e²1 (- + The general solution is x(1) = x₂ (1) + x₂ (1) =Ge² The particular solution is x, q₁ +2a₂ = 0 sint-cost cost 2 --(²) - cost - sin t sin t sint) 051) + C₂0²³ ( 1)+(²1) ² -cost-sint sint Help with this one Find the unique solution to the inhomogeneous system above which satisfies x(0) -
Transcribed Image Text:Consider the system of differential equations Find the general solution to this system, given that the general solution to the corresponding homo- geneous system is x₁ (t) = 3x₁ (t) + 2x₂(t) x₂ (t) = x1(t) + x₂(t) + e²t xh(t) = C₁e²t (sin x₂ (1)= ae²¹ = (a ₁) ²² Given the system of differential equations x' (1)=3x₁ (1)+2x₂ (1) x₂² (1) = -x₂₁ (1) + x₂ (1) e²¹ and the homogeneous solution is sint-cost -cost-sint x₂ (1)=C₁e²¹ +C₂e²¹ cost sint The objective is to find the general solution to the system. Find the particular solution. By using the method of undetermined coefficients, the particular solution is of the form Take derivative. x,' (1)=(201₁) e²¹ Plug the values into the given system. 3 2 a₁ (29)² =(³, 1)(a)~²¹ +(2₂) -1 3a,e² +2a₂e² -a₁e²² + a₂e²¹ +e²₁ → t - cost cos t Equating the coefficients. 2a₁ = 3a₁ +2a₂ 2a₂ = -a₁ + a₂ +1 a₁ + a₂ = 1 Solving both equations gives a, = 2 and a₂ =-1. st) + C₂e²1 (- + The general solution is x(1) = x₂ (1) + x₂ (1) =Ge² The particular solution is x, q₁ +2a₂ = 0 sint-cost cost 2 --(²) - cost - sin t sin t sint) 051) + C₂0²³ ( 1)+(²1) ² -cost-sint sint Help with this one Find the unique solution to the inhomogeneous system above which satisfies x(0) -
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