Solving \( \sqrt{1 - Y^2} \,\mathrm{d}x = (\sin^{-1} Y - X) \,\mathrm{d}y \) A Step-by-Step Guide

Introduction

In this article, we will delve into the process of solving the differential equation ${ \sqrt{1 - y^2} \,\mathrm{d}x = (\sin^{-1} y - x) \,\mathrm{d}y }$. This equation presents an interesting challenge, requiring a combination of algebraic manipulation and integration techniques. Our approach will involve rearranging the equation, recognizing its structure, and applying an appropriate method to find the general solution. Differential equations like this appear in various fields of science and engineering, making their solutions highly relevant and practical. Understanding the steps involved in solving such equations not only enhances mathematical proficiency but also provides valuable problem-solving skills applicable in broader contexts. The solution process will be detailed, ensuring clarity and comprehension for readers with a basic understanding of calculus and differential equations.

Understanding the Problem

To effectively solve the differential equation, we must first understand its nature and structure. The given equation is ${ \sqrt{1 - y^2} \,\mathrm{d}x = (\sin^{-1} y - x) \,\mathrm{d}y }$. It involves a relationship between ${ x }$ and ${ y }$ and their differentials, ${ \mathrm{d}x }$ and ${ \mathrm{d}y }$. This suggests that it is a first-order differential equation. Our initial step is to rearrange the terms to bring the equation into a more recognizable form. Dividing both sides by ${ \mathrm{d}y }$ and ${ \sqrt{1 - y^2} }$, we get:

${ \frac{\mathrm{d}x}{\mathrm{d}y} = \frac{\sin^{-1} y - x}{\sqrt{1 - y^2}} }$

This form is still not immediately solvable, but it allows us to see the equation's structure more clearly. We can further rearrange it to isolate ${ \frac{\mathrm{d}x}{\mathrm{d}y} }$ and ${ x }$ terms on one side:

${ \frac{\mathrm{d}x}{\mathrm{d}y} + \frac{x}{\sqrt{1 - y^2}} = \frac{\sin^{-1} y}{\sqrt{1 - y^2}} }$

This rearranged form reveals the equation as a first-order linear differential equation in terms of ${ x }$ as a function of ${ y }$. Recognizing this structure is crucial because it guides us to the appropriate solution method: the integrating factor technique. The integrating factor technique is a powerful method for solving first-order linear differential equations, and its applicability here stems from the equation's adherence to the standard form. This recognition step is critical in the problem-solving process, allowing us to apply a known method efficiently. By identifying the equation type, we can avoid trial-and-error approaches and proceed with a systematic solution.

Rearranging the Equation

As highlighted earlier, rearranging the given differential equation is a pivotal step in our solution process. Starting from the original equation, ${ \sqrt{1 - y^2} \,\mathrm{d}x = (\sin^{-1} y - x) \,\mathrm{d}y }$, we aim to isolate the derivative term and structure the equation into a standard form that facilitates easier solving. The initial manipulation involves dividing both sides by ${ \mathrm{d}y }$ and ${ \sqrt{1 - y^2} }$, yielding:

${ \frac{\mathrm{d}x}{\mathrm{d}y} = \frac{\sin^{-1} y - x}{\sqrt{1 - y^2}} }$

This step is essential because it expresses the rate of change of ${ x }$ with respect to ${ y }$, which is a fundamental aspect of differential equations. However, to apply standard solution techniques, we need to further rearrange the terms. The next step involves moving the ${ x }$ term from the right side to the left side, resulting in:

${ \frac{\mathrm{d}x}{\mathrm{d}y} + \frac{x}{\sqrt{1 - y^2}} = \frac{\sin^{-1} y}{\sqrt{1 - y^2}} }$

This form is now recognizable as a first-order linear differential equation. A first-order linear differential equation has the general form ${ \frac{\mathrm{d}x}{\mathrm{d}y} + P(y)x = Q(y) }$, where ${ P(y) }$ and ${ Q(y) }$ are functions of ${ y }$. Comparing this general form with our rearranged equation, we can identify:

${ P(y) = \frac{1}{\sqrt{1 - y^2}} }$

${ Q(y) = \frac{\sin^{-1} y}{\sqrt{1 - y^2}} }$

These identifications are crucial for the subsequent steps in solving the equation, particularly when applying the integrating factor method. By recognizing the equation's form, we can strategically apply the appropriate techniques and avoid common pitfalls. This systematic approach is essential for solving complex differential equations effectively. The rearranged equation sets the stage for the next phase, where we will compute the integrating factor and proceed with the integration process.

Identifying the Integrating Factor

Having rearranged the differential equation into the standard first-order linear form, the next crucial step is to identify the integrating factor. The integrating factor, denoted as ${ \mu(y) }$, is a function that, when multiplied by the differential equation, makes the left-hand side an exact derivative. This allows us to integrate both sides and find the solution. For a first-order linear differential equation of the form ${ \frac{\mathrm{d}x}{\mathrm{d}y} + P(y)x = Q(y) }$, the integrating factor is given by:

${ \mu(y) = e^{\int P(y) \,\mathrm{d}y} }$

In our case, we have identified ${ P(y) = \frac{1}{\sqrt{1 - y^2}} }$. Therefore, we need to compute the integral of ${ P(y) }$ with respect to ${ y }$:

${ \int P(y) \,\mathrm{d}y = \int \frac{1}{\sqrt{1 - y^2}} \,\mathrm{d}y }$

This integral is a standard form, and its solution is the inverse sine function, ${ \sin^{-1} y }$. Thus,

${ \int \frac{1}{\sqrt{1 - y^2}} \,\mathrm{d}y = \sin^{-1} y }$

Now, we can find the integrating factor ${ \mu(y) }$ by exponentiating the result:

${ \mu(y) = e^{\sin^{-1} y} }$

The integrating factor ${ \mu(y) = e^{\sin^{-1} y} }$ is a critical component in solving the differential equation. This function, when multiplied across the equation, transforms the left-hand side into a derivative that can be easily integrated. The significance of the integrating factor lies in its ability to simplify the integration process, making an otherwise complex problem solvable. With the integrating factor identified, the next step involves multiplying it by the entire differential equation and then integrating both sides to find the general solution. This systematic approach is essential for solving first-order linear differential equations, ensuring accuracy and efficiency in the solution process.

Multiplying by the Integrating Factor

With the integrating factor ${ \mu(y) = e^{\sin^{-1} y} }$ determined, the next step is to multiply both sides of the rearranged differential equation by this factor. The rearranged equation is:

${ \frac{\mathrm{d}x}{\mathrm{d}y} + \frac{x}{\sqrt{1 - y^2}} = \frac{\sin^{-1} y}{\sqrt{1 - y^2}} }$

Multiplying both sides by ${ e^{\sin^{-1} y} }$ gives us:

${ e^{\sin^{-1} y} \frac{\mathrm{d}x}{\mathrm{d}y} + e^{\sin^{-1} y} \frac{x}{\sqrt{1 - y^2}} = e^{\sin^{-1} y} \frac{\sin^{-1} y}{\sqrt{1 - y^2}} }$

The purpose of multiplying by the integrating factor is to transform the left-hand side of the equation into the derivative of a product. Specifically, we aim to show that the left-hand side is the derivative of ${ x \cdot e^{\sin^{-1} y} }$ with respect to ${ y }$. To verify this, we apply the product rule for differentiation:

${ \frac{\mathrm{d}}{\mathrm{d}y} \left( x e^{\sin^{-1} y} \right) = \frac{\mathrm{d}x}{\mathrm{d}y} e^{\sin^{-1} y} + x \frac{\mathrm{d}}{\mathrm{d}y} \left( e^{\sin^{-1} y} \right) }$

Using the chain rule, we differentiate ${ e^{\sin^{-1} y} }$ with respect to ${ y }$:

${ \frac{\mathrm{d}}{\mathrm{d}y} \left( e^{\sin^{-1} y} \right) = e^{\sin^{-1} y} \frac{\mathrm{d}}{\mathrm{d}y} \left( \sin^{-1} y \right) = e^{\sin^{-1} y} \frac{1}{\sqrt{1 - y^2}} }$

Substituting this back into the product rule, we get:

${ \frac{\mathrm{d}}{\mathrm{d}y} \left( x e^{\sin^{-1} y} \right) = e^{\sin^{-1} y} \frac{\mathrm{d}x}{\mathrm{d}y} + x e^{\sin^{-1} y} \frac{1}{\sqrt{1 - y^2}} }$

This is exactly the left-hand side of our multiplied equation. Therefore, we can rewrite the equation as:

${ \frac{\mathrm{d}}{\mathrm{d}y} \left( x e^{\sin^{-1} y} \right) = e^{\sin^{-1} y} \frac{\sin^{-1} y}{\sqrt{1 - y^2}} }$

This transformation is the key to solving the differential equation. By recognizing the left-hand side as a derivative, we can now integrate both sides with respect to ${ y }$ to find the solution. The next step involves performing this integration, which will lead us to the general solution of the differential equation.

Integrating Both Sides

After multiplying by the integrating factor and recognizing the left-hand side as a derivative, we have the equation:

${ \frac{\mathrm{d}}{\mathrm{d}y} \left( x e^{\sin^{-1} y} \right) = e^{\sin^{-1} y} \frac{\sin^{-1} y}{\sqrt{1 - y^2}} }$

To proceed, we integrate both sides with respect to ${ y }$. The integration of the left-hand side is straightforward, as it is the integral of a derivative:

${ \int \frac{\mathrm{d}}{\mathrm{d}y} \left( x e^{\sin^{-1} y} \right) \,\mathrm{d}y = x e^{\sin^{-1} y} + C_1 }$

where ${ C_1 }$ is the constant of integration. Now, we need to integrate the right-hand side:

${ \int e^{\sin^{-1} y} \frac{\sin^{-1} y}{\sqrt{1 - y^2}} \,\mathrm{d}y }$

This integral requires a substitution to simplify it. Let ${ u = \sin^{-1} y }$. Then, ${ \frac{\mathrm{d}u}{\mathrm{d}y} = \frac{1}{\sqrt{1 - y^2}} }$, and ${ \mathrm{d}u = \frac{\mathrm{d}y}{\sqrt{1 - y^2}} }$. Substituting these into the integral, we get:

${ \int e^u u \,\mathrm{d}u }$

This integral can be solved using integration by parts. Integration by parts states that ${ \int a \,\mathrm{d}b = ab - \int b \,\mathrm{d}a }$. Let ${ a = u }$ and ${ \mathrm{d}b = e^u \,\mathrm{d}u }$. Then, ${ \mathrm{d}a = \mathrm{d}u }$ and ${ b = e^u }$. Applying integration by parts:

${ \int e^u u \,\mathrm{d}u = ue^u - \int e^u \,\mathrm{d}u = ue^u - e^u + C_2 }$

Substituting back ${ u = \sin^{-1} y }$, we have:

${ \int e^{\sin^{-1} y} \frac{\sin^{-1} y}{\sqrt{1 - y^2}} \,\mathrm{d}y = \sin^{-1} y \cdot e^{\sin^{-1} y} - e^{\sin^{-1} y} + C_2 }$

Now, equating the integrated left-hand side and right-hand side, we get:

${ x e^{\sin^{-1} y} = \sin^{-1} y \cdot e^{\sin^{-1} y} - e^{\sin^{-1} y} + C }$

where ${ C = C_2 - C_1 }$ is the combined constant of integration. This equation represents the general solution to the given differential equation. The final step involves simplifying this expression to obtain a more concise form of the solution.

Simplifying the Solution

We have arrived at the equation:

${ x e^{\sin^{-1} y} = \sin^{-1} y \cdot e^{\sin^{-1} y} - e^{\sin^{-1} y} + C }$

To simplify this expression, we can divide both sides by ${ e^{\sin^{-1} y} }$. This gives us:

${ x = \sin^{-1} y - 1 + \frac{C}{e^{\sin^{-1} y}} }$

We can rewrite the constant term as ${ Ce^{-\sin^{-1} y} }$, where ${ C }$ is an arbitrary constant. Thus, the simplified solution is:

${ x = \sin^{-1} y - 1 + Ce^{-\sin^{-1} y} }$

This equation represents the general solution to the differential equation ${ \sqrt{1 - y^2} \,\mathrm{d}x = (\sin^{-1} y - x) \,\mathrm{d}y }$. The solution expresses ${ x }$ in terms of ${ y }$ and includes an arbitrary constant ${ C }$, which accounts for the family of solutions that satisfy the differential equation. This constant can be determined if we have an initial condition, such as a specific value of ${ x }$ for a given value of ${ y }$.

The simplified solution provides a clear and concise representation of the relationship between ${ x }$ and ${ y }$ that satisfies the original differential equation. It is important to note that this solution is valid for ${ -1 < y < 1 }$, as the original equation involves ${ \sqrt{1 - y^2} }$ and ${ \sin^{-1} y }$, both of which have restrictions on the domain of ${ y }$. This final solution encapsulates the entire process of solving the differential equation, from rearranging the terms and identifying the integrating factor to performing the integration and simplifying the result. It highlights the power of systematic problem-solving techniques in differential equations and provides a valuable tool for analyzing systems described by such equations.

Conclusion

In conclusion, we have successfully solved the differential equation ${ \sqrt{1 - y^2} \,\mathrm{d}x = (\sin^{-1} y - x) \,\mathrm{d}y }$. The process involved several key steps, including rearranging the equation into a first-order linear form, identifying and applying the integrating factor, integrating both sides, and simplifying the resulting expression. The final solution, ${ x = \sin^{-1} y - 1 + Ce^{-\sin^{-1} y} }$, represents a family of solutions parameterized by the constant ${ C }$. This solution is valid for ${ -1 < y < 1 }$, considering the domain restrictions imposed by the original equation.

The journey through this problem highlights the importance of recognizing the structure of a differential equation and applying appropriate solution techniques. The integrating factor method proved to be a powerful tool in this context, allowing us to transform the equation into an integrable form. This approach is widely applicable to other first-order linear differential equations, making it a valuable technique in the field of mathematics and its applications. Solving differential equations is a fundamental skill in various areas of science and engineering, as these equations often model real-world phenomena. The detailed step-by-step solution provided in this article serves as a comprehensive guide for students and professionals alike, enhancing their problem-solving abilities and deepening their understanding of differential equations.