Question

The solution of the difference equation \((\mathrm{dy} / \mathrm{dx})=\left[1 /\left\\{\mathrm{xy}\left(\mathrm{x}^{2} \sin \mathrm{y}^{2}+1\right)\right\\}\right]\) is: (A) \(x^{2}\left(\cos y^{2}-\sin y^{2}-e^{-(y) 2}\right)=4\) (B) \(y^{2}\left(\cos y^{2}-\sin y^{2}-2 c e^{-(y) 2}\right)=2\) (C) \(x^{2}\left(\cos y^{2}-\sin y^{2}-2 c e^{-(y) 2}\right)=2\) (D) none of these

Short answer

The solution of the difference equation \(\frac{dy}{dx} = \frac{1}{xy(x^2\sin y^2 + 1)}\) is: (D) none of these

Step-by-step solution

Separate the variables

We need to express the given differential equation in the form \(g(y) dy = h(x) dx\). To do this, we can rewrite the given equation as: \(dy = \frac{1}{y(x^2\sin y^2 + 1)} dx\) Now, the variables are separated.

Integrate both sides

Next, we integrate both sides with respect to their respective variables: \(\int dy = \int \frac{1}{y(x^2\sin y^2 + 1)} dx\) On the left side, the integral is straightforward: \(\int dy = y + C_1\) On the right side, since the integral is not elementary, let's denote it by a function F(x): \(\int \frac{1}{y(x^2\sin y^2 + 1)} dx = F(x) + C_2\)

Combine the constants and express the solution

We can combine the constants on both sides by writing \(C = C_1 - C_2\), leading to the general solution: \(y + C = F(x)\) But this first-order equation cannot be expressed in elementary functions, which prevents us from solving it completely.

Compare the result with the provided options

Since we cannot integrate without any simplification or substitution, we can infer from this that none of the options match the form we have above. Therefore, the answer is: (D) none of these

Key concepts

Variable Separation

To solve a differential equation through variable separation, we express all terms involving one variable on one side of the equation and all terms involving the other variable on the opposite side. This approach is often used with first-order differential equations, making the equation easier to integrate.

In the exercise provided, our initial task was to rearrange the equation into a form where each variable was kept on separate sides. Starting with the original equation, we wanted it to resemble the format: \(g(y) dy = h(x) dx\).

We achieved this by isolating \(dy\) on one side and rearranging the remaining terms to accompany \(dx\) on the other, thus forming:

• The left side became \(dy\), representing the \(y\) component of the integration.
• The right side transformed into \(\frac{1}{y(x^2\sin y^2 + 1)} dx\), capturing the \(x\)-dependent aspects.

After completing this separation, each side was set up perfectly for the next step, which involves integration. Variable separation doesn't solve the equation entirely but simplifies it into an integrable form.

Integration Techniques

Once the variables are separated, the next step is to integrate each side with respect to its respective variable. This is a fundamental process in solving differential equations.

In our exercise, the integration on the left-hand side, \(\int dy\), was quite simple and resulted in \(y + C_1\). However, the right-side integration, \(\int \frac{1}{y(x^2\sin y^2 + 1)} dx\), posed a challenge since it's not straightforward to integrate.

In situations where direct integration is difficult, we represent the integral as a function of \(x\), denoted here as \(F(x)\). This step is crucial as it acknowledges that, without advanced techniques or substitutions, we can't find an explicit elementary function for this integration.

Thus, our equation transformed into:

• \(y + C_1 = F(x) + C_2\)

Finally, the constants from both sides were combined to simply \(C = C_1 - C_2\), culminating in our generalized expression. Despite not yielding a complete solution, this integration step is essential for progressing towards a potential solution.

First-Order Differential Equations

First-order differential equations involve functions and their first derivatives and are generally represented in the standard form. These equations are foundational in many applications, ranging from physics to economics. Our exercise is an example of such an equation.

With first-order differential equations, the primary goal is to find a function that satisfies the equation. This often involves integration after separating the variables, as previously discussed.

While solving first-order equations, especially non-linear or complex ones like ours, it may not be always possible to express the solution in terms of elementary functions directly. Often, these equations require:

• Partial solutions or representations in terms of known functions, or
• Approximations using numerical methods.

In our case, after we separated and integrated, our first-order differential equation couldn't be solved easily for an explicit function in elementary terms. This is common with complex equations, indicating that sometimes the end of the process is recognizing the limitations of the direct methods, as we concluded with option (D), 'none of these.'

First-order differential equations, despite their simple order, can still present significant challenges, ensuring that problem-solving skills and creativity are essential in finding solutions.