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Problem 2

# Show that if $$e \neq 0$$ and $$\alpha>1, \beta>0$$, then no solution of $$\ddot{\varphi}(t)+\beta \dot{\varphi}(t)+(\alpha-1) \varphi(t)=e$$

Expert verified
Under the given conditions $$e \neq 0$$, $$\alpha > 1$$, and $$\beta > 0$$, there is no particular solution to the second-order linear inhomogeneous differential equation $$\ddot{\varphi}(t) + \beta \dot{\varphi}(t) + (\alpha - 1) \varphi(t) = e$$. Thus, this differential equation has no solution under these conditions.
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## Step 1: Identify the given equation

We are given the following second-order linear inhomogeneous differential equation: $\ddot{\varphi}(t) + \beta \dot{\varphi}(t) + (\alpha - 1) \varphi(t) = e,$ where $$\ddot{\varphi}(t)$$ represents the second derivative of $$\varphi$$ with respect to time $$t$$, $$\dot{\varphi}(t)$$ represents the first derivative, and $$\alpha$$, $$\beta$$, and $$e$$ are constants with $$e \neq 0$$, $$\alpha > 1$$, and $$\beta > 0$$.

## Step 2: Look for a particular solution

To solve this inhomogeneous differential equation, we look for a particular solution $$\varphi_p(t)$$. Since the right-hand side of the equation is a constant, we can try to find a particular solution in the form of another constant, say $$\varphi_p(t) = C$$, where $$C$$ is a constant.

## Step 3: Plug the particular solution into the equation

Now, let's plug $$\varphi_p(t) = C$$ into the given differential equation. The first derivate $$\dot{\varphi}(t)$$ and the second derivate $$\ddot{\varphi}(t)$$ of the constant function $$\varphi_p(t) = C$$ are both zero: $0 + \beta \cdot 0 + (\alpha - 1) C = e.$ Simplifying, we have: $(\alpha - 1) C = e.$

## Step 4: Analyze the results

Now, we can see that the only term left on the left-hand side of the equation is $$(\alpha - 1) C$$. For this equation to have a solution for $$C$$, we must have $$(\alpha - 1) C = e \neq 0$$. However, as $$\alpha > 1$$, we can't find any constant $$C$$ that satisfies the equation $$(\alpha - 1) C = 0$$. Because of that, under the given conditions, no particular solution $$\varphi_p(t)$$ in the form of a constant can be found for this differential equation.

## Step 5: Conclusion

Given the conditions $$e \neq 0$$, $$\alpha > 1$$, and $$\beta > 0$$, we have shown that there is no particular solution to the second-order linear inhomogeneous differential equation: $\ddot{\varphi}(t) + \beta \dot{\varphi}(t) + (\alpha - 1) \varphi(t) = e.$ Therefore, the differential equation has no solution under these conditions.

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