Question

In Exercises \(15-18,\) solve the differential equation. $$G^{\prime}(t)=\frac{2 t^{3}}{t^{3}-t}$$

Short answer

The solution to the given differential equation is \(G(t) = ± \sqrt{t^{4} + C}\) where C is the constant of integration.

Step-by-step solution

Rewrite the Differential Equation

We can rewrite the differential equation by bringing the variables on each side of the equation as follows: \((t^{3}-t) dG = 2 t^{3} dt\).

Integrate Both Sides

We now have to integrate both sides of the equation. We get: \(\int (t^{3}-t) dG = \int 2 t^{3} dt\). The integral of the left hand side can be computed using power rule, resulting in \(\frac{1}{4}G^{4}-\frac{1}{2}G^{2} + C\) while the integral on the right-hand side is computed as \(\frac{1}{2}t^{4}\), where C is the constant of integration.

Solve for G(t)

We know that: \(\frac{1}{4}G^{4}-\frac{1}{2}G^{2} + C = \frac{1}{2}t^{4}\). Now we solve for \(G(t)\). To make this easy, we can let \(u= G^{2}\), so the equation becomes quadratic: \(\frac{1}{2}u^{2}-u+C = \frac{1}{2}t^{4}\). Solving this quadratic equation, we find that \(u = G^{2}\) is equal to \(t^{4} + C\). We can then take the square root of both sides to obtain the function \(G(t) = ± \sqrt{t^{4} + C}\), where the ± sign accounts for the two possible roots of the square, namely the positive and the negative root.

Key concepts

Integration

Integration is a fundamental concept in calculus, central to solving differential equations. It involves finding a function when its rate of change, or derivative, is known. In the context of differential equations, integrating both sides of an equation is akin to accumulating the change to retrieve the original function.

For example, in the exercise provided, once the differential equation \(G'(t) = \frac{2t^3}{t^3 - t}\) is written in a separable form, you can integrate each side of the equation separately. Performing the integration process leads to finding the antiderivative of each function, which adds a constant of integration (usually denoted as \(C\)) to one side. This constant represents the indefinite nature of the integral, as it can take on any value.For students: Remember, the goal of integration in solving differential equations is to recover the original function from its derivative. It's like piecing together the whole story from just knowing how it progresses over time.

Separation of Variables

Separation of Variables is a method used to solve a differential equation by separating the variables involved onto different sides of the equation. It's like untangling two sets of strings, so that each set can be dealt with individually.

Take the exercise at hand. The differential equation \(G'(t) = \frac{2t^3}{t^3 - t}\) initially looks like a complex tangle. The first step is to rewrite it into a form where each side of the equation contains only one variable: \((t^3 - t) dG = 2 t^3 dt\). This allows us to handle each side independently through integration, as both sides are now 'separated'.

Tips for Success

• Check for common factors that you can cancel out before separating variables.
• Don't forget to include the differential (\(dt\) or \(dG\)) when you separate the variables.
• After separation, integrate both sides carefully with respect to their own variables.

For students: It’s crucial to understand that not all differential equations can be solved using separation of variables, but when they can, this method is a powerful and straightforward tool.

Power Rule

The Power Rule is a useful technique for both differentiation and integration, particularly when dealing with polynomials. It states that the integral of \(x^n\), where \(n\) is any real number except -1, is \(\frac{x^{n+1}}{n+1}\), plus the constant of integration, \(C\).

In the Exercise

The given exercise requires integrating \((t^3 - t) dG\) and \((2t^3 dt)\), which feature terms that are perfect candidates for the Power Rule. For the right side, \((2t^3 dt)\), using the Power Rule, we integrate and increase the exponent by one, leading to \(\frac{2}{4}t^4\), or \(\frac{1}{2}t^4\). Similarly, when applied to a term like \(G^2\) when solving for \(G(t)\), we use the reverse of the power rule in the process of finding the antiderivative.For students: Mastering the Power Rule can significantly simplify the process of finding integrals of polynomial functions. Just remember to always add the constant of integration at the end of the process, and adjust the exponent appropriately.