Question

Solve the given differential equation. $$ x y^{\prime}=\left(1-y^{2}\right)^{1 / 2} $$

Short answer

Question: Solve the first-order differential equation $xy' = (1-y^2)^{1/2}$, and find the general solution. Answer: The general solution for the given first-order differential equation is $y=\sin(\ln|x|+C)$.

Step-by-step solution

Separate the variables

Divide both sides by \(x(1-y^2)^{1/2}\) to separate the variables: $$ \frac{y'}{(1-y^2)^{1/2}}=\frac{1}{x} $$

Integrate both sides

Integrate both sides with respect to x: $$ \int\frac{y'}{(1-y^2)^{1/2}}dx=\int\frac{1}{x}dx $$ Now, let's integrate both sides. For the left side, we notice that the integral is an arctangent integral. For the right side, the integral results in a natural logarithm.

Perform the integration

We'll integrate the left side using arctangent and the right side using natural logarithm: $$ \int\frac{y'}{(1-y^2)^{1/2}}dx=\arcsin(y) + C_1 $$ $$ \int\frac{1}{x}dx=\ln|x| + C_2 $$ Now, we'll combine the constants of integration (C_1 and C_2) into one constant, C: $$ \arcsin(y) - \ln|x| = C $$

Solve for y

Now, let's solve for y to obtain an explicit formulation for our solution: $$ y=\sin(\ln|x|+C) $$ This is the general solution for the given first-order differential equation.

Key concepts

Separation of Variables

Separation of Variables is a straightforward technique used to solve differential equations. The primary goal is to rearrange the equation such that each variable and its differential are on different sides of the equation.

In the example given, we started with the differential equation: \[ x y^{\prime} = \left(1-y^{2}\right)^{1/2} \] By dividing both sides of the equation by \( x(1-y^2)^{1/2} \), we successfully separated the variables, isolating terms involving \( y \) on one side and terms involving \( x \) on the other: \[ \frac{y'}{(1-y^2)^{1/2}} = \frac{1}{x} \] The trick is to arrange terms such that each variable can be integrated independently. This tool is particularly efficient when dealing with equations where direct integration is possible once separation is achieved.

Integration

Integration plays a crucial role in solving differential equations. Once a differential equation is separated into integrable components, the next task is to find antiderivatives.

In our case, we've transformed the original differential equation into: \[ \int \frac{y^\prime}{\sqrt{1-y^2}} \, dx = \int \frac{1}{x} \, dx \] The left-hand side can be recognized as a form that integrates into an arcsine function, while the right-hand side integrates into a natural logarithm. Integration for both sides is straightforward once they are in standard integrable forms:

• The left-hand side: \( \int \frac{dy}{\sqrt{1-y^2}} \) is \( \arcsin(y) \).
• The right-hand side: \( \int \frac{1}{x}dx \) is \( \ln|x| \).

Therefore, integration provides us with an interconnection between both expressions necessary for further steps toward solving the differential equation.

Arcsine Function

The Arcsine Function is a crucial component in solving certain types of differential equations, especially when the integral involves expressions of the form \( \frac{1}{\sqrt{1-y^2}} \).

In this exercise, noticing this form allows us to use the inverse trigonometric function, specifically, arcsine: \[ \int \frac{dy}{\sqrt{1-y^2}} = \arcsin(y) + C_1 \] The arcsine function, \( \arcsin(y) \), is the inverse of the sine function and ranges between -\( \frac{\pi}{2} \) and \( \frac{\pi}{2} \).

Recognizing patterns that resolve into such trigonometric inverses can greatly simplify the task of integration and lead to a concise form of the solution for the differential equation.

General Solution

The General Solution is the expression that incorporates all solutions of a differential equation. It typically includes arbitrary constants stemming from the integration process.

Once we've completed the integration in the exercise, we derive the following form: \[ \arcsin(y) - \ln|x| = C \] This expression holds for any permissible value of \( y \) and \( x \), giving a family of solutions which are encapsulated by the constant \( C \).

Converting this implicit form to an explicit form for \( y \), we derive the general solution: \[ y = \sin(\ln|x| + C) \] This step elucidates the behavior of solutions across different values of \( x \) and indicates how they are controlled by the constant \( C \). The general solution is critical in understanding the complete solution space of the differential equation and how particular solutions can be extracted based on specific initial conditions or constraints.