To demonstrate that $x(t) \to \pm \infty$ in finite time for the differential equation

$\frac{dx}{dt} = rx + x^3$

with $r > 0$, and any initial condition $x_0 \neq 0$, we proceed by analyzing the behavior of solutions over time.

Step 1: Analyze the structure of the differential equation

We have a first-order nonlinear differential equation

$\frac{dx}{dt} = rx + x^3$

Here, $r > 0$ is a constant. Notice that the equation contains a linear term $rx$ and a cubic nonlinearity $x^3$.

Step 2: Separate variables

We will use separation of variables to better understand the time behavior. Rewrite the equation as:

$\frac{1}{rx + x^3} \, dx = dt$

This allows us to express time $t$ as an integral of $x$:

$t - t_0 = \int_{x_0}^{x(t)} \frac{1}{rx + x^3} \, dx$

Step 3: Simplify the integrand

Factor the denominator as follows:

$rx + x^3 = x(r + x^2)$

Thus, the integral becomes:

$t - t_0 = \int_{x_0}^{x(t)} \frac{1}{x(r + x^2)} \, dx$

This integral becomes difficult to evaluate exactly, but we can analyze its behavior as $x \to \infty$.

Step 4: Behavior of the solution for large $x$

For large $x$, the cubic term $x^3$ dominates the linear term $rx$, so for large values of $x$, the differential equation approximates:

$\frac{dx}{dt} \approx x^3$

Solving this simpler equation:

$\frac{dx}{x^3} = dt$

Integrating both sides:

$-\frac{1}{2x^2} = t + C$

Solving for $x$, we get:

$x(t) = \pm \frac{1}{\sqrt{2(C - t)}}$

From this, it's clear that as $t \to C$, $x(t) \to \pm \infty$, which shows that $x(t) \to \pm \infty$ in finite time.

Step 5: Conclusion

Since the large $x$ behavior dominates and leads to $x(t) \to \pm \infty$ in finite time, we conclude that for any initial condition $x_0 \neq 0$, the solution will blow up to infinity (either positive or negative) in finite time.

Thus, starting from any initial condition $x_0 \neq 0$, the solution $x(t)$ will tend to $\pm \infty$ in finite time.

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Would you like more details or clarification on any specific part?

Related questions:

1. How does the behavior of the solution change for $r < 0$?
2. Can we calculate the exact time at which $x(t)$ blows up?
3. How does the initial condition $x_0$ influence the time of blow-up?
4. What is the role of the cubic term $x^3$ in determining the blow-up time?
5. What happens if $x_0 = 0$?

Tip:

For equations with cubic nonlinearities, the large $x$ behavior often dominates, leading to blow-up or other extreme behaviors in finite time. Always check this limit to understand the global behavior of solutions.