Question

Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. $$y^{\prime}(t) e^{t / 2}=y^{2}+4$$

Short answer

Based on the step-by-step solution above, create a short answer question: Question: Given the first-order differential equation \(y^{\prime}(t) e^{t / 2}=y^{2}+4\), find the general solution in terms of the independent variable t. Answer: The general solution is \(y(t) = 2\tan{(-4e^{-t/2} + C)}\), where C is a constant.

Step-by-step solution

Rearrange the equation

We can start by dividing both sides of the equation by \(e^{t/2}\), in order to get: $$y^{\prime}(t) = (y^{2}+4)e^{-t/2}$$ Now, we have a separable equation.

Use separation of variables

We need to separate the variables to each side of the equation: $$\frac{dy}{dt} = (y^{2} + 4)e^{-t/2}$$ Dividing both sides by \((y^{2} + 4)\) and multiplying by \(dt\) yields: $$\frac{dy}{y^{2} + 4} = e^{-t/2} dt$$ Now, we integrate both sides:

Integrate

Integrate both sides with respect to their respective variables: $$\int \frac{dy}{y^2 + 4} = \int e^{-t/2} dt$$ On the left side, we have a simple rational function, which can be integrated using a substitution or by recognizing that it is an arctangent form. The left side integral evaluates to: $$\frac{1}{2}\arctan \frac{y}{2} + C_1$$ On the right side, we have an exponential function that can easily be integrated: $$-2e^{-t/2} + C_2$$ Now we set both sides equal: $$\frac{1}{2}\arctan \frac{y}{2} + C_1 = -2e^{-t/2} + C_2$$

Solve for y

We want to solve for y as a function of t, so we need to perform some algebraic manipulations: First, subtract \(C_1\) from both sides: $$\frac{1}{2}\arctan \frac{y}{2} = -2e^{-t/2} + (C_2 - C_1)$$ Now, to remove the arctangent, we take the tangent of both sides: $$\frac{y}{2} = \tan{(-4e^{-t/2} + (C_2 - C_1))}$$ Finally, multiply both sides by 2 to get y explicitly: $$y(t) = 2\tan{(-4e^{-t/2} + (C_2 - C_1))}$$ So the final general solution for the given differential equation, expressed explicitly as a function of the independent variable, is: $$y(t) = 2\tan{(-4e^{-t/2} + C)}$$ where \(C = C_2 - C_1\) is a constant.

Key concepts

Separation of Variables

The method of separation of variables is a powerful technique for solving differential equations. It allows you to split the equation into parts where each variable is on a different side.
This way, you can integrate each side individually. In the problem provided, we have an equation involving a derivative, a function of \( y \), and a function of \( t \).
By manipulating and rearranging terms, we separate \( y \) and its differential \( dy \) on one side, and \( t \) along with its differential \( dt \) on the other:

• Left Side: \( \frac{dy}{y^2 + 4} \)
• Right Side: \( e^{-t/2} dt \)

This separation prepares the expression for integration, simplifying the process of finding the function \( y(t) \). This method is especially useful for first-order ordinary differential equations (ODEs).

General Solution

Finding the general solution is the aim when solving a differential equation. It encompasses all possible solutions of the equation. In the context of our exercise, the general solution of a differential equation involves integrating both sides after performing the initial separation of variables.
The end goal is to express \( y \) explicitly as \( y(t) \), which indicates finding relationships between \( y \) and \( t \) that satisfy the initial differential equation.
During the integration step, the constants of integration, usually noted as \( C \), play a crucial role. They depict the generality of the solution.
The final expression in our problem, \( y(t) = 2\tan(-4e^{-t/2} + C) \), highlights the dependent-variable function \( y \) related through an independent variable \( t \), encompassing an arbitrary constant \( C \).
This constant allows the solution to satisfy a range of initial conditions.

Integration Techniques

Integration is the process of finding an antiderivative, and it's used to solve the separated parts of the differential equation. In this exercise, two different integrals emerge:

• \( \int \frac{dy}{y^2 + 4} \) represents a rational function that can often be approached using a substitution method or recognizing known formulae like the arctangent.
• \( \int e^{-t/2} dt \) involves an exponential function, which integrates into another exponential form with adjusted coefficients.

On the left, the integration of \( \frac{dy}{y^2 + 4} \) results in \( \frac{1}{2} \arctan \frac{y}{2} \), utilizing arctangent’s properties. On the right, integration turns \( e^{-t/2} \) into \( -2e^{-t/2} \), following standard exponential integration rules.
These techniques help unravel the components of variables leading to a General Solution that satisfies the original differential equation.