Question

Solve the following initial value problems. When possible, give the solution as an explicit function of \(t\) $$y^{\prime}(t)=\frac{\cos ^{2} t}{2 y}, y(0)=-2$$

Short answer

Based on the given initial value problem and following a step-by-step solution, the explicit function of t, y(t), is found to be: $$y(t) = -\sqrt{\frac{1}{2}(t+\frac{1}{2}\sin(2t))+4}$$

Step-by-step solution

Separate Variables

We start by rewriting the given differential equation in a separable form. Separate the variables by multiplying both sides by \(2y\) and dividing both sides by \(\cos^2 t\): $$2y \cdot y'(t) = \cos^2 t$$ Now, rewrite the equation in terms of differentials: $$2y \, dy = \cos^2 t \, dt$$

Integrate Both Sides

Now, integrate both sides of the equation with respect to their respective variables: $$\int 2y \, dy = \int \cos^2 t \, dt$$ On the left side, we have: $$y^2 = \int \cos^2 t \, dt + C_1$$ On the right side, let's use the power-reduction formula \(\cos^2 t = \frac{1 + \cos(2t)}{2}\): $$y^2 = \int\frac{1+\cos(2t)}{2}\, dt + C_1 = \frac{1}{2}\int(1+\cos(2t))\, dt$$ Now integrate: $$y^2 = \frac{1}{2}(t + \frac{1}{2}\sin(2t)) + C_1$$

Apply the Initial Condition

Apply the initial condition \(y(0)=-2\) to find the constant \(C_1\): $$(-2)^2 = \frac{1}{2}(0 + \frac{1}{2}\sin(0)) + C_1$$ Solving for \(C_1\), we get: $$C_1 = 4$$ So the equation becomes: $$y^2= \frac{1}{2}(t+\frac{1}{2}\sin(2t))+4$$

Solve for y(t)

Now, solve for \(y(t)\): $$y(t) = \pm\sqrt{\frac{1}{2}(t+\frac{1}{2}\sin(2t))+4}$$ However, since \(y(0) = -2\), we take the negative square root to satisfy the initial condition: $$y(t) = -\sqrt{\frac{1}{2}(t+\frac{1}{2}\sin(2t))+4}$$ Therefore, the explicit solution to the given initial value problem is: $$y(t) = -\sqrt{\frac{1}{2}(t+\frac{1}{2}\sin(2t))+4}$$

Key concepts

Understanding Differential Equations

Differential equations are mathematical equations that relate some function with its derivatives. They play a crucial role in fields such as physics, engineering, and economics, as they can model real-world systems and predict their behavior over time.

The equation given in our problem, \(y^{\prime}(t)=\frac{\cos ^{2} t}{2 y}\), is a type known as a first-order differential equation because it involves the first derivative of the function \(y\) with respect to time \(t\).

First-order differential equations often appear with initial conditions, like \(y(0)=-2\) in this case, which specify the value of the function at a particular point. Solving this means finding a function \(y(t)\) that satisfies both the differential equation and the initial condition.

• **Ordinary Differential Equations**: Involves functions of a single variable and its derivatives.
• **Partial Differential Equations**: Involves functions of multiple variables and their partial derivatives.

Differential equations can model everything from population growth to electrical circuits, underlining their vast importance in both theoretical studies and practical applications.

Separation of Variables Technique

Solving a differential equation often involves finding ways to isolate the variables involved, giving rise to the method of "separation of variables." This method is particularly handy for solving separable equations, which are those that can be arranged such that each side of the equation depends only on one variable.

In our problem, by multiplying through by \(2y\) and dividing by \(\cos^2 t\), we successfully separated the variables:
\[ 2y \cdot y'(t) = \cos^2 t \]
Then it became easy to integrate each side with respect to its own variable:
\[ 2y \, dy = \cos^2 t \, dt \]

This separation allows you to handle each integral independently. It simplifies the process to solve differential equations step-by-step:

• **Identify**: Can the equation be rearranged so that each side depends only on one variable?
• **Rearrange**: Separate the equation into two parts.
• **Integrate**: Solve each part independently through integration.

The ability to separate and then reintegrate variables makes this method particularly intuitive and effective for numerous differential equations.

Integration Techniques in Differential Equations

Integrating both sides of a separated differential equation allows us to find the general solution to an initial value problem. Integration can sometimes be straightforward, but often requires clever techniques or substitutions to simplify the integral.

In the given exercise, the right side of the equation needed a special approach to integrate \(\cos^2 t\). By utilizing a trigonometric identity, \(\cos^2 t = \frac{1 + \cos(2t)}{2}\), we simplified the integration process:
\[ \int \cos^2 t \, dt = \int \frac{1 + \cos(2t)}{2} \, dt \]
This transformation into a more manageable form allowed us to integrate term by term:
\[ \frac{1}{2}(t + \frac{1}{2}\sin(2t)) + C_1 \]

Without such techniques, certain integrals could remain unsolvable using simple antiderivatives. Techniques often used in such cases include:

• **Substitution**: To rearrange the integral into an easier form.
• **Parts**: A method involving the product of functions.
• **Trigonometric Identities**: Useful for simplifying integrals involving trigonometric functions.

Mastering these integration techniques helps significantly in finding solutions to complex differential equations effectively and efficiently.