Question

Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. $$t^{-3} y^{\prime}(t)=1$$

Short answer

Question: Given the first-order ordinary differential equation (ODE) $$t^{-3} y^{\prime}(t)=1$$, find the general solution. Answer: The general solution for the given ODE is $$y(t) = \frac{t^4}{4} + C$$, where C is an arbitrary constant.

Step-by-step solution

Write the ODE in standard form

We need to write the given ODE in the standard form of first-order ODEs: $$y^{\prime}(t) = F(t)$$ To do this, multiply both sides of the equation by \(t^3\): $$t^{3}\cdot t^{-3} y^{\prime}(t) = t^3 \cdot 1$$ This gives us: $$y^{\prime}(t) = t^3$$

Integrate both sides of the equation

Integrate both sides of the equation with respect to the independent variable t: $$\int y^{\prime}(t) dt = \int t^3 dt$$

Find the antiderivative

The antiderivative of \(y^{\prime}(t)\) with respect to t is \(y(t)\). The antiderivative of \(t^3\) with respect to t can be found using the power rule: $$\int t^{3} dt = \frac{t^{4}}{4}+C$$

Find the general solution

Combine the results from Step 3 to find the general solution of the ODE: $$y(t) = \frac{t^{4}}{4}+C$$ Where C is an arbitrary constant. This is the general solution of the given ODE, expressed explicitly as a function of the independent variable t.

Key concepts

First-Order Ordinary Differential Equations

A first-order ordinary differential equation (ODE) involves a function and its first derivative. These equations represent rates of change and are commonly used in fields like physics, engineering, and biology. The standard form of a first-order ODE can be expressed as \( y'(t) = F(t, y) \), where \( y'(t) \) is the derivative of \( y \) with respect to \( t \), and \( F(t, y) \) is a given function of both the independent variable \( t \) and the dependent variable \( y \). In our problem, we initially have \( t^{-3} y'(t) = 1 \). By multiplying through by \( t^3 \), we rearrange it into the form \( y'(t) = t^3 \), which is a clear first-order ODE. Recognizing this form helps us utilize integration as the next step for solving the equation.

• Linear ODEs: A first-order ODE is called linear if it can be expressed as \( y'(t) + p(t)y = q(t) \).
• Separable ODEs: If \( F(t, y) \) can be written as the product of separate functions of \( t \) and \( y \), it can be solved by separation of variables.

Antiderivatives

Antiderivatives are functions that reverse differentiation. For a given function \( f(t) \), an antiderivative is a function \( F(t) \) such that \( F'(t) = f(t) \). Antiderivatives are an essential tool for solving differential equations, as they allow us to recover the original function from its derivative.
The antiderivative is closely related to the integral. For example, when we solve \( y'(t) = t^3 \), we are looking for an antiderivative for \( t^3 \). Using the power rule, the antiderivative of \( t^n \) is \( \frac{t^{n+1}}{n+1} \). Hence, the antiderivative of \( t^3 \) is \( \frac{t^4}{4} \). By identifying antiderivatives, we can progress towards finding the general solution of a differential equation.

Integration

Integration is the mathematical process of finding antiderivatives and is a core method used to solve differential equations. Specifically, indefinite integration or finding an antiderivative plays a crucial role. It is represented by the integral symbol \( \int \), and the result includes an arbitrary constant \( C \). This constant signifies all the possible antiderivatives.
When we integrate \( y'(t) = t^3 \), we set up the integral \( \int y'(t) dt = \int t^3 dt \). This operation yields \( y(t) = \frac{t^4}{4} + C \). Here are a few key points:

• Definite vs. Indefinite Integration: Definite integration calculates the area under the curve between two points. Indefinite integration finds a general form without boundaries.
• Rules of Integration: Familiarity with integral rules, such as the power rule, constant rule, and more, is vital for solving with ease.

Understanding integration allows us to translate rates of change back into original expressions.

General Solution

The general solution of a differential equation is a family of solutions that incorporates all possible solutions, typically including an arbitrary constant. For the ODE \( y'(t) = t^3 \), after integration, we arrive at \( y(t) = \frac{t^4}{4} + C \). The constant \( C \) represents the infinite number of parallel curves, and each specific value of \( C \) corresponds to a particular solution.
To find a particular solution, additional conditions, known as initial values or boundary conditions, are often provided. By substituting these conditions into the general solution, we can determine the specific solution that fits that scenario. In summary, the general solution provides a comprehensive view of all potential behaviors of the system described by the differential equation.