We are given the differential equation:

$\frac{dx}{dt} = -x + 2e^t$

with the initial condition $x(0) = 2$.

Step 1: Solve the Differential Equation

This is a linear first-order differential equation. We can solve it using the integrating factor method.

The equation is in the form:

$\frac{dx}{dt} + x = 2e^t$

1. The integrating factor (IF) is given by $e^{\int 1 \, dt} = e^t$.

2. Multiply both sides of the equation by $e^t$: $e^t \frac{dx}{dt} + e^t x = 2e^{2t}$

3. The left-hand side is now the derivative of $e^t x$, so: $\frac{d}{dt} \left( e^t x \right) = 2e^{2t}$

4. Integrating both sides with respect to $t$: $e^t x = \int 2e^{2t} \, dt = e^{2t} + C$

5. Solving for $x$: $x = e^t + C e^{-t}$

Step 2: Apply Initial Condition

We know that $x(0) = 2$. Substitute $t = 0$ and $x(0) = 2$ into the equation: $2 = e^0 + C e^0$ $2 = 1 + C$ $C = 1$

Thus, the solution to the differential equation is: $x(t) = e^t + e^{-t}$

Step 3: Check the Given Conditions

Now, we will check the conditions $A$, $B$, $C$, $D$, and $E$.

1. A) $x(\ln 2) = \frac{5}{2}$

Substitute $t = \ln 2$ into the solution: $x(\ln 2) = e^{\ln 2} + e^{-\ln 2} = 2 + \frac{1}{2} = \frac{5}{2}$

So, A is true.

2. B) $x(-\ln 5) = \frac{26}{5}$

Substitute $t = -\ln 5$ into the solution: $x(-\ln 5) = e^{-\ln 5} + e^{\ln 5} = \frac{1}{5} + 5 = \frac{26}{5}$

So, B is true.

3. C) $x(\ln 5) \neq \frac{26}{5}$

Substitute $t = \ln 5$ into the solution: $x(\ln 5) = e^{\ln 5} + e^{-\ln 5} = 5 + \frac{1}{5} = \frac{26}{5}$

So, C is false.

4. D) $x(\ln 5) \neq \frac{26}{5}$

As shown above, $x(\ln 5) = \frac{26}{5}$, so D is false.

5. E) $x(-\ln 5) \neq \frac{26}{5}$

From our earlier calculation, we know that $x(-\ln 5) = \frac{26}{5}$, so E is false.

Final Answer:

• A is true.
• B is true.
• C is false.
• D is false.
• E is false.

Would you like further details on any of the steps, or do you have other questions?

Here are 5 related questions to deepen your understanding:

1. How does the integrating factor method work in solving linear differential equations?
2. Can you show an example of a non-homogeneous differential equation and how it’s solved?
3. What is the physical interpretation of the solution $x(t) = e^t + e^{-t}$ in terms of growth and decay?
4. What happens if we change the initial condition to $x(0) = 5$? How would the solution change?
5. How do you find the general solution of a non-linear first-order differential equation?

Tip: Always check the initial conditions and simplify exponential expressions carefully to avoid computational mistakes.