Question

Suppose
y
=
2
e
−
4
x
y=2e−4x
is the solution to the initial value problem
y
′
+
k
y
=
0
,
y
(
0
)
=
y
0
y′+ky=0,y(0)=y0
. Find the value of
y
0

Answer 1

Answer on Question #69469 – Math – Differential Equations

Question

1. Suppose $y = 2e^{-4x}$ is the solution to initial value problem $y' + ky = 0, y(0) = y_0$. Find the value of $y_0$.

Solution

1. Construct a characteristic equation:

$$t + k = 0; \, t = -k.$$

2. The general solution of the differential equation:

$$y = C e^{-kx}.$$

3. We know that $y(0) = y_0$:

$$y(0) = C e^0 = C$$

4. So $y(0) = C$, $y = 2e^{-4x}$ is the solution to initial value problem. Therefore, $y_0 = 2$.

Answer: $y_0 = 2$.

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