Suppose
𝑦
1
(
𝑡
)
=
2
𝑒
5
𝑡
and
𝑦
2
(
𝑡
)
=
1
2
𝑒
−
𝑡
are solutions of a second-order homogeneous linear ODE on
𝑅
. Which one of the following is also a solution to the same ODE?

Question

Suppose  
𝑦
1
(
𝑡
)
=
2
𝑒
5
𝑡
 and   
𝑦
2
(
𝑡
)
=
1
2
𝑒
−
𝑡
 are solutions of a second-order homogeneous linear ODE on  
𝑅
. Which one of the following is also a solution to the same ODE?

BlackTom AI · Accepted Answer

We are told that y1(t) = 2 e^{5t} and y2(t) = (1/2) e^{-t} are solutions of a second-order homogeneous linear ODE with real variable t on R. For such an ODE, any linear combination of its solutions is also a solution, so the set of all solutions contains all expressions of the form C1 y1(t) + C2 y2(t).

Option 1: y(t) = e^{5t} − 2
This is not a linear combination of y1 and y2, because it lacks the e^{5t} term with the correct coefficient (it would require a term proportional to e^{5t}, not a constant  -2). Since constants by themselves do not arise from a linear combination of y1 and y2, this is not generally a solution.

Option 2: y(t) = e^{t} + e^{-t}
This is a sum of e^{t} and e^{-t}. However, neither e^{t} nor e^{-t} alone is given as a multiple of y1 or y2, and unless the ODE’s characteristic roots were exactly r = 1 and r = -1 (which would force e^{t} and e^{-t} as basis solutions), this combination is unlikely to be a solution. Given we know the specific basis y1 and y2, this option does not align with their linear span.

Option 3: y(t) = e^{5t} + e^{t}
Similar to Option 2, this is a sum of exponentials with bases that do not correspond to the known solutions y1 and y2. It cannot be expressed as a linear combination of y1 and y2 with constant coefficients unless e^{t} is itself in the span of y1 and y2, which we have no basis for. Therefore this is not guaranteed to be a solution.

Option 4: y(t) = 2 e^{5t} − (1/2) e^{t}
This option includes a term 2 e^{5t}, which matches a multiple of y1 (since y1 = 2 e^{5t}), and a second term involving e^{t}. While the second term e^{t} is not directly given by y2, notice that any linear combination of y1 and y2 has the form (a)(2 e^{5t}) + (b)( (1/2) e^{-t} ) = (2a) e^{5t} + (b/2) e^{-t}. If we want the second term to be (−1/2) e^{t}, we would need a term involving e^{t} rather than e^{-t}; however, our expression for a general solution uses e^{5t} and e^{−t} terms, not e^{t}. That said, the exact combination 2 e^{5t} − (1/2) e^{t} does not appear to be directly formed from y1 and y2 unless the ODE’s solution set spans e^{t} as well. Yet, since the problem statement asserts these two given functions are solutions, any linear combination of them must be a solution. If we attempt to express y(t) = 2 e^{5t} − (1/2) e^{t} as α y1 + β y2, we would have α(2 e^{5t}) + β( (1/2) e^{-t} ) = (2α) e^{5t} + (β/2) e^{-t}. To match 2 e^{5t} − (1/2) e^{t}, we would need a term proportional to e^{t} which cannot be produced from this span. Therefore, as written, this option cannot be represented as a linear combination of the given solutions, and is not generally a solution.

Summary: Among the given options, the only one that can be obtained as a linear combination of y1 and y2 is the one that matches their span. Since y1 = 2 e^{5t} and y2 = (1/2) e^{-t}, the general solution is C1 y1 + C2 y2 = 2 C1 e^{5t} + (C2/2) e^{-t}. The expression 2 e^{5t} − (1/2) e^{−t} corresponds to C1 = 1 and C2 = −1, giving 2 e^{5t} − e^{−t}, which differs in the coefficient of the e^{−t} term from −(1/2) e^{−t}. If we strictly require a combination that matches the given y2’s coefficient, we would have −1 e^{−t} rather than −(1/2) e^{−t}. However, the key idea is that the correct form must lie in the linear span of y1 and y2. Between the options, only a form that aligns with this span is acceptable as a solution; the presented “2 e^{5t} − (1/2) e^{−t}” is the one that fits the expected pattern of combining the two known solutions with some coefficient choices, whereas the other options fail to lie in the span of {y1, y2}.

MATHS 208 Quiz 30

Suppose 𝑦 1 ( 𝑡 ) = 2 𝑒 5 𝑡 and 𝑦 2 ( 𝑡 ) = 1 2 𝑒 − 𝑡 are solutions of a second-order homogeneous linear ODE on 𝑅 . Which one of the following is also a solution to the same ODE?

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