Question

Finding general solutions Find the general solution of each differential equation. Use \(C, C_{1}, C_{2}, \ldots\) to denote arbitrary constants. $$y^{\prime}(t)=12 t^{5}-20 t^{4}+2-6 t^{-2}$$

Short answer

Question: Find the general solution of the following differential equation: $$\frac{dy}{dt} = 12t^5 - 20t^4 + 2 - 6t^{-2}$$ Solution: The general solution of the given differential equation is: $$y(t) = 2t^6 - 4t^5 + 2t + 6t^{-1} + C$$

Step-by-step solution

Rewrite the given differential equation

Start by rewriting the given equation as: $$\frac{dy}{dt} = 12t^5 - 20t^4 + 2 - 6t^{-2}$$

Integrate both sides with respect to t

Now integrate both sides with respect to t: $$\int \frac{dy}{dt} \, dt = \int (12t^5 - 20t^4 + 2 - 6t^{-2}) \, dt$$

Perform the integration on the right side

Integrate each term on the right side separately: $$\int \frac{dy}{dt} \, dt = 12\int t^5 \, dt - 20\int t^4 \, dt + \int 2 \, dt - 6\int t^{-2} \, dt$$ Use the power rule for integration for each term, which states that \(\int t^n \, dt = \frac{t^{n+1}}{n+1} + C\): $$y(t) = 12\frac{t^6}{6} - 20\frac{t^5}{5} + 2t - 6\frac{t^{-1}}{-1} + C$$

Simplify the general solution

Simplify the expression to get the general solution of the differential equation: $$y(t) = 2t^6 - 4t^5 + 2t + 6t^{-1} + C$$ Now we have the general solution of the differential equation with an arbitrary constant C.

Key concepts

Differential Equation Integration

When it comes to solving differential equations, the integration process is a key tool. A differential equation involves an unknown function and its derivative(s). In our exercise, we are given a differential equation:\[ \frac{dy}{dt} = 12t^5 - 20t^4 + 2 - 6t^{-2} \]This means we need to find the original function \( y(t) \) whose derivative with respect to \( t \) yields the right-hand side of the equation. To tackle this, we perform integration.The process involves integrating each term of the equation with respect to \( t \), effectively reversing the differentiation. This results in the general solution, which represents a family of functions that satisfy the differential equation. The main task is to integrate the terms on the right, combining our results to write out a solution for \( y(t) \). This process gives us the template for finding solutions to different forms of differential equations.

Power Rule for Integration

The power rule for integration is a fundamental tool when solving integrals, especially in solving differential equations like the one in our exercise. The rule is straightforward: it lets you integrate any term of the form \( t^n \) where \( n \) is any real number except \( -1 \).For an expression \( \int t^n \, dt \), the integral is given by:\[ \int t^n \, dt = \frac{t^{n+1}}{n+1} + C \]Here, \( C \) is the constant of integration, which accounts for the family of solutions.In our specific problem, each term on the right-hand side of the equation can be integrated using the power rule:- For \( 12t^5 \), use \( n = 5 \) to obtain \( 12 \frac{t^{6}}{6} \).- For \( -20t^4 \), use \( n = 4 \) to get \( -20 \frac{t^{5}}{5} \).- The constant \( 2 \) integrates simply to \( 2t \).- For \( -6t^{-2} \), use \( n = -2 \) to find \( -6 \frac{t^{-1}}{-1} \).After applying the power rule, visualize integration as unwinding the differentiation. Always remember, the rule supplies a shortcut for obtaining antiderivatives with ease.

Arbitrary Constants in Integration

When integrating to solve differential equations, we encounter arbitrary constants. These constants are crucial because they signify that our solution represents a family of functions.After performing integration, each integrated term yields a part of the solution, plus a constant denoted here as \( C \). The general solution illustrated in our exercise is:\[ y(t) = 2t^6 - 4t^5 + 2t + 6t^{-1} + C \]Here, \( C \) is the arbitrary constant and plays an important role in defining an entire set of solutions. Since differential equations describe processes that can have multiple conditions or initial values, the arbitrary constant allows for flexibility and specificity depending on additional conditions.When you solve differential equations, knowing how to handle these constants is essential. They allow you to adjust solutions according to given initial values or boundary conditions, ultimately helping you tailor solutions to specific scenarios.