Determine if the given function y = f(x) is a solution of the accompanying differential equation.
-Differential equation: $\mathrm { y } ^ { \prime } = \mathrm { e } ^ { - 8 \mathrm { x } } - 8 \mathrm { y }$
Initial condition: $y ( - 8 ) = 0$
Solution candidate: $y = ( x + 8 ) e ^ { - 8 x }$

Question

Quizplus · Accepted Answer

To determine whether the solution candidate $y = ( x + 8 ) e ^ { - 8 x }$ is a solution of the differential equation $\mathrm { y } ^ { \prime } = \mathrm { e } ^ { - 8 \mathrm { x } } - 8 \mathrm { y }$ with initial condition $y ( - 8 ) = 0$, we need to substitute $y$ into the differential equation and check if it satisfies both the differential equation and the initial condition.

First, let's calculate the derivative of $y$ with respect to $x$:
$$\frac { d y } { d x } = e ^ { - 8 x } \cdot \frac { d } { d x } ( x + 8 ) - ( x + 8 ) \cdot \frac { d } { d x } ( e ^ { - 8 x } )$$

Using the product rule and chain rule, we get:
$$\frac { d y } { d x } = e ^ { - 8 x } - 8 ( x + 8 ) e ^ { - 8 x }$$

Now, let's substitute $y$ and $\frac { d y } { d x }$ into the differential equation:
$$e ^ { - 8 x } - 8 ( x + 8 ) e ^ { - 8 x } = e ^ { - 8 x } - 8 ( x + 8 ) e ^ { - 8 x }$$

We can see that this is true, so the solution candidate $y = ( x + 8 ) e ^ { - 8 x }$ does indeed satisfy the differential equation.

Finally, we need to check if the solution candidate satisfies the initial condition $y ( - 8 ) = 0$. We get:
$$y ( - 8 ) = ( - 8 + 8 ) e ^ { - 8 ( - 8 ) } = 0$$

Since the solution candidate satisfies both the differential equation and the initial condition, we can conclude that the answer is A: True.

Determine If the Given Function Y = F(x) Is a Solution