Given the separable differential equation
𝑦
′
−
3
𝑡
3
𝑦
=
0
and the initial condition
𝑦
(
0
)
=
6
. Find the explicit solution to the differential (answer should be in y=f(t) format.)

Question

Given the separable differential equation 
𝑦
′
−
3
𝑡
3
𝑦
=
0
 and the initial condition 
𝑦
(
0
)
=
6
. Find the explicit solution to the differential  (answer should be in y=f(t) format.)

BlackTom AI · Accepted Answer

We start by identifying the given differential equation and initial condition to guide our reasoning about the solution.

Option 1: y = e^{[3 t 4 4]} + 5. This form resembles an exponential solution but with a additive constant 5, which cannot satisfy a homogeneous linear equation y' - 3 t^3 y = 0 under any initial condition. The equation is homogeneous, so the solution should be a constant multiple of the exponential factor, not a plus constant term. Hence this option is incorrect.

Option 2: y = 6 e 6 t 2. This expression is garbled and does not conform to the standard exponential form y = C e^{f(t)}. Moreover, the exponent written as 6 t 2 is not a meaningful function of t for the differential equation, and would not satisfy the differential equation or initial condition. Therefore this option is incorrect.

Option 3: y = 6 e [ 3 t 4 4 ]. Interpreting the bracketed expression as the exponent 3 t^4 / 4, this matches the standard solution form for a first-order linear ODE with integrating factor: y' - 3 t^3 y = 0 has integrating factor μ(t) = exp(-∫3 t^3 dt) = exp(- (3/4) t^4), leading to y = C exp((3/4) t^4). Applying the initial condition y(0) = 6 gives C = 6, so y = 6 exp((3/4) t^4). This aligns with the option’s structure and the initial condition, so this option is correct.

Option 4: y = e [ 3 t 4 4 ] + 6. This is similar to Option 1 but with only a single plus 6 instead of 5; however, adding a constant to a homogeneous solution does not yield a valid solution to y' - 3 t^3 y = 0 with the given initial condition. The correct solution is a constant multiple of the exponential, not an exponential plus a constant. Therefore this option is incorrect.

In summary, only Option 3 reproduces the correct exponential form with the constant determined by the initial condition, while the other options fail due to inappropriate constant terms, garbled exponents, or incorrect additive structure.

MATH_1026_17235_202501 1026 - Practice Test II- Requires Respondus LockDown Browser

Given the separable differential equation 𝑦 ′ − 3 𝑡 3 𝑦 = 0 and the initial condition 𝑦 ( 0 ) = 6 . Find the explicit solution to the differential (answer should be in y=f(t) format.)

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