Question

Solving initial value problems Solve the following initial value problems. $$y^{\prime}(x)=4 \sec ^{2} 2 x, y(0)=8$$

Short answer

Question: Determine the function y(x) that satisfies the initial value problem \(y'(x) = 4\sec^2 2x\) and \(y(0) = 8\). Answer: The function y(x) that satisfies the given initial value problem is \(y(x) = 2\tan 2x + 8\).

Step-by-step solution

Integrate the given derivative

In order to find the function, we will integrate the given derivative with respect to x. $$\int y'(x) dx = \int 4\sec^2 2x dx$$ We will use the substitution method for integration by letting: $$u = 2x$$ Then, the differential of u is: $$du = 2 dx$$ So, we can rewrite the integral as follows: $$\frac{1}{2}\int 4\sec^2 u du$$ Now, we can integrate by recalling that the integral of \(\sec^2 u\) is equal to the tangent function: $$\frac{1}{2} (4\int\sec^2 u du) = 2\tan u + C$$ Now, we will substitute back for x: $$2\tan 2x + C$$

Use the initial value to determine the value of C

The initial value provided is \(y(0) = 8\), so we will plug in x = 0 into our function and set it equal to 8: $$8 = 2\tan (2\cdot 0) + C$$ Since \(\tan 0 = 0\), we find that the constant C is equal to 8: $$C = 8$$

Write the final solution

Now, we can write the final solution for the initial value problem: $$y(x) = 2\tan 2x + 8$$

Key concepts

Integration Techniques

When you encounter an integral in calculus, there are various techniques available to find its solution. These techniques include the basic rule of integration, integration by parts, integration by partial fractions, and substitution method. The choice of technique largely depends on the form of the function being integrated.

For instance, when the integrand is a product of functions, integration by parts might be useful. If we have a complex rational expression, partial fractions can break it down into simpler pieces. On the other hand, the substitution method is particularly helpful when dealing with integrands that include composite functions—functions within functions—that can be simplified through a change of variable.

Understanding when and how to apply these techniques will significantly ease the process of integration. By mastering these methods, you'll be able to tackle a wide range of integrals, whether they appear in straightforward exercises or real-world applications.

Secant Squared Function

The secant squared function, written as \(\sec^2 x\), is fundamental in calculus, especially when working with trigonometric integrals. It is the square of the secant function, which itself is the reciprocal of the cosine function: \(\sec x = \frac{1}{\cos x}\).

What makes \(\sec^2 x\) significant is its direct relationship with the tangent function through the derivative. The derivative of \(\tan x\) is \(\sec^2 x\), a fact that is often used in reverse for integration. Knowing this relationship allows us to integrate any function involving \(\sec^2 x\) directly, returning a tangent function plus a constant of integration. This is a cornerstone concept that demonstrates the interplay between differentiation and integration in trigonometric functions.

Substitution Method for Integration

Substitution is a powerful tool for simplifying integrals and is often compared to the 'change of variables' in algebra. When dealing with composite functions that may not be immediately integrable, applying substitution can transform the integral into a more manageable form.

The process involves identifying a part of the integrand as a new variable—let's call it \(u\). We then find the differential of \(u\), denoted \(du\), which replaces a corresponding part of the original integrand in terms of \(dx\). This prepares the integral for direct integration with respect to \(u\) instead of \(x\).

After integrating with respect to \(u\), the final step is to substitute back in terms of \(x\), reversing the initial substitution step. With substitution, integrals that initially seem insurmountable can be systematically broken down into simpler forms that lead to a solution.