Differential Equations

Step 1: Given Differential Equation
We are given the first-order linear differential equation: $ \alpha \beta y_{\alpha, \beta}(x) S = \{ y_{\alpha, \beta}(x) : \alpha, \beta \in \mathbb{R} \} e^{\alpha x} e^{\alpha x} y x u = x du = dx dv = e^{(\beta + \alpha) x} dx v = \frac{e^{(\beta + \alpha) x}}{\beta + \alpha} y y(1) = 1 x = 1 C \alpha \beta C y_{\alpha, \beta}(x) \frac{df}{dx} \alpha \beta S \alpha \beta S f(x) = \frac{x^2}{2} e^{-x} + \left(e - \frac{1}{2}\right) e^{-x} f(x) = \frac{e^x}{2} \left(x - \frac{1}{2}\right) + \left(e - \frac{e^2}{4}\right) e^{-x} S$.

First-Order Separable Differential Equation

Given:

The differential equation is:

Rearranging the equation to solve for :

Now, we have a first-order separable differential equation. Let's separate the variables:

Next, we integrate both sides:

Integrating both expressions will give us the function .

Given: The differential equation is:

Step 1: Rearranging the Equation

We start by rearranging the equation as follows:

The integrating factor is:

Step 2: Solving the Differential Equation

Now, we solve the differential equation:

Substituting , we find:

Step 3: Final Solution

The solution to the differential equation is:

Step 4: Function Behavior

Given the form of the solution, is an even function. The solution is split into two parts:

• If , then or .
• If , then or .

Step 5: Determining the Increasing Function

For an increasing function, we check when the derivative . Solving this gives:

The function is increasing over the interval .

Step 6: Graph and Maximum Value

We plot the graph of and observe the following points:

• At and , the value is 16.
• At , the value is 15.

The maximum value of is 16, and the function reaches its maximum at .

Solution of the Given Differential Equation

Given: The equation is:

Step 1: Rewriting the equation

We start by rewriting the equation as:

Step 2: Solving the left-hand side integral

Now we solve the left-hand side integral:

We can factor the denominator and use partial fraction decomposition to solve the integral:

Step 3: Simplifying the equation

We can simplify the equation as follows:

Step 4: Analyzing the equation for possible solutions

We now solve for in terms of and analyze the possible cases.

Case 1: If , we get:

Substituting gives us a rejected solution since it leads to an undefined value.

Case 2: If , we get:

Step 5: Calculating specific values

We can now calculate specific values for :

Next, we compute :

Now, calculate the final answer:

Conclusion

The final value of is 8, hence the answer is .

To solve the problem, we need to analyze the given limit condition and find the function satisfying it.

1. Understanding the limit expression:
Given:
for each .

2. Rewrite the limit to use the derivative concept:
Notice the numerator and denominator both tend to zero as . This suggests using the definition of derivative or applying algebraic manipulation.

3. Rewrite numerator and denominator:

4. Separate the fraction:

5. Analyze the first fraction as :
Since , using derivative formulas:
Both numerator and denominator tend to zero, apply L'Hôpital's Rule:
Derivative of numerator w.r.t. :
Derivative of denominator w.r.t. :
Thus, the limit is:

6. Analyze the second fraction as :

7. Substitute limits back:

8. Multiply both sides by 9:

9. Rearranged differential equation:
or

10. Divide both sides by :
This is a linear differential equation of the form:
with and .

11. Find the integrating factor (IF):

12. Multiply entire equation by integrating factor:
which simplifies to

13. Integrate both sides:

14. Solve for :

15. Use the initial condition :

Final form of :