Find the function y(t) that satisfies the differential equation
dy

dt
−7t6y=24t2et7
and the condition y(0)=−4.

Question

Find the function y(t) that satisfies the differential equation
dy

dt
−7t6y=24t2et7
and the condition y(0)=−4.

BlackTom AI · Accepted Answer

We start with the given differential equation: dy/dt − 7 t^6 y = 24 t^2 e^{t^7}, with the initial condition y(0) = −4.

First, identify the integrating factor for the linear first-order ODE. The coefficient of y is −7 t^6, so the integrating factor μ(t) is exp(−∫7 t^6 dt) = exp(−t^7).

Multiply every term by μ(t): e^{−t^7} dy/dt − 7 t^6 e^{−t^7} y = 24 t^2 e^{t^7} e^{−t^7}.

The left-hand side becomes the derivative of (y e^{−t^7}) with respect to t: d/dt [ y e^{−t^7} ] = 24 t^2.

Integrate both sides with respect to t: y e^{−t^7} = ∫ 24 t^2 dt = 8 t^3 + C, where C is the constant of integration.

Solve for y: y(t) = (8 t^3 + C) e^{t^7}.

Apply the initial condition y(0) = −4: y(0) = (0 + C) e^{0} = C = −4. So the particular solution is y(t) = (8 t^3 − 4) e^{t^7}.

Now compare with the options:
- Option 1: y(t) = 8 t^3 e^{t^7}. This misses the correct constant term, so it does not satisfy y(0) = −4.
- Option 2: y(t) = (8 t^3 + 4) e^{t^7}. The sign of the constant is incorrect, and it would give y(0) = 4, not −4.
- Option 3: y(t) = (t^7 + 24 t^6 − 4) e^{t^7}. This form is not compatible with the derived solution and would not yield the correct derivative structure.
- Option 4: y(t) = (8 t^3 − 4) e^{t^7}. This matches the computed particular solution and satisfies y(0) = −4.

Thus, the correct function is y(t) = (8 t^3 − 4) e^{t^7}, corresponding to the fourth option.

MATH1062/1005/1023 (ND) MATH1062/1023 Calculus Quiz 4

Find the function y(t) that satisfies the differential equation dy dt −7t6y=24t2et7 and the condition y(0)=−4.

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