Question

Solve the given differential equation. $$ y^{\prime}=\left(\cos ^{2} x\right)\left(\cos ^{2} 2 y\right) $$

Short answer

Question: Find the solution to the given differential equation: \(y'(x) = \cos^2(x)\cos^2(2y)\). Answer: The solution to the given differential equation is: \(y = \frac{1}{2}\arctan\left(x + \frac{1}{2}\sin(2x) + 2C\right)\), where C is the constant of integration.

Step-by-step solution

Separate Variables

Divide both sides of the equation by \(\cos^2(2y)\) to separate variables: $$ \frac{y^{\prime}}{\cos^2(2y)} = \cos^2(x) $$

Integrate Both Sides

Integrate both sides of the equation with respect to their respective variables: $$ \int{\frac{1}{\cos^2(2y)} \mathrm{d}y^{\prime}}= \int {\cos^2(x) \mathrm{d}x} $$

Solve the Integral

Solving the integral, we get: $$ \int{\sec^2(2y) \mathrm{d}y} = \int {\cos^2(x) \mathrm{d}x} $$ On the left side, we can use the property: \(\int \sec^2(ax) \, dx = \frac{1}{a} \tan(ax) + C_1\). Substituting \(a = 2\), we get: $$ \frac{1}{2} \tan(2y) = \int {\cos^2(x) \mathrm{d}x} $$ To determine the integral of \(\cos^2(x)\), we will use the power reduction formula: \(\cos^2(x) = \frac{1}{2}(1 + \cos(2x))\). Now, we have: $$ \frac{1}{2} \tan(2y) = \int {\frac{1}{2}(1 + \cos(2x)) \mathrm{d}x} $$ Integrating the right side, we get: $$ \frac{1}{2} \tan(2y) = \frac{1}{2} \left(x + \frac{1}{2}\sin(2x)\right) + C $$ where \(C = C_1 - C_2\)

Solve for 'y'

Solve for 'y' in terms of 'x' to obtain the final solution: $$ \tan(2y) = x + \frac{1}{2}\sin(2x) + 2C $$ $$ 2y = \arctan\left(x + \frac{1}{2}\sin(2x) + 2C\right) $$ $$ y = \frac{1}{2}\arctan\left(x + \frac{1}{2}\sin(2x) + 2C\right) $$ So, the solution to the given differential equation is: $$ y = \frac{1}{2}\arctan\left(x + \frac{1}{2}\sin(2x) + 2C\right) $$

Key concepts

Separation of Variables

Separation of Variables is a fundamental technique in solving differential equations. It involves rearranging a differential equation so that each variable and its differential are isolated on opposite sides of the equation. This step allows you to integrate each side independently, making the equation simpler to solve.

To apply this technique, identify terms involving the dependent variable (in our case, 'y') and move them to one side of the equation. Similarly, move terms involving the independent variable (here, 'x') to the other side. For example, in the exercise, we started with the equation:

• \( y^{\prime} = \left(\cos^{2}(x)\right)\left(\cos^{2}(2y)\right) \)
• We divided both sides by \( \cos^{2}(2y) \), and rearranged it to: \( \frac{y^{\prime}}{\cos^{2}(2y)} = \cos^{2}(x) \)

Effectively using Separation of Variables requires that the new form of the equation, achieved from the original, is ready to be integrated. The clarity of your separations strengthens accuracy and ensures that each part can be integrated with respect to its relevant variable.

Integration Techniques

Integration is the process of finding a function whose derivative matches a given function, and it is a crucial step in solving differential equations split by separation of variables. Different integration techniques help compute the integral as efficiently and accurately as possible.

In our exercise, after separating the variables, we had:

• \( \int \sec^2(2y) \, dy = \int \cos^2(x) \, dx \)

For the left side involving \( \sec^2(2y) \), we can use a straightforward property of the secant function, such that:

• \( \int \sec^2(ax) \, dx = \frac{1}{a} \tan(ax) + C_1 \)

Whereas, on the right side, we employed the power reduction formula to express the cosine squared function more simply:

• \( \cos^2(x) = \frac{1}{2}(1 + \cos(2x)) \)

This transformed the integral into a format suitable for direct integration. Understanding and choosing the right integration technique simplifies your work and is the key to solving otherwise complex expressions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all variable values. They are extremely useful for simplifying trigonometric expressions and solving equations. In the exercise, these identities played a critical role in both rewriting and integrating the functions.

Initially, we used the identity:

• \( \cos^2(x) = \frac{1}{2}(1 + \cos(2x)) \)

This identity came into play when integrating on the right side. The identity helps to break down \( \cos^2(x) \) into two simpler components: a constant and another cosine function, making integration more manageable.

Understanding identities such as this allows one to deal with seemingly complex trigonometric expressions with increased ease. They are not just valuable tools for integration, but also for rearranging and reducing equations in a variety of mathematical and applied contexts.