Problem 21 - Advanced Engineering Mathematics

Question

In Problems 1-22, solve the given differential equation by separation of variables. $$ \frac{d y}{d x}=x \sqrt{1-y^{2}} $$

Step-by-Step Solution

Answer

The general solution is \( y = \sin \left( \frac{x^2}{2} + C \right) \).

Step 1: Identify variables and rewrite the equation

The given differential equation is \( \frac{dy}{dx} = x \sqrt{1-y^2} \). This indicates that we need to separate the variables \( x \) and \( y \) on each side of the equation.

Step 2: Rearrange the terms

Rearrange the differential equation to separate the \( y \) terms from the \( x \) terms. This gives us \( \frac{dy}{\sqrt{1-y^2}} = x \, dx \).

Step 3: Integrate both sides

Now, integrate both sides of the equation. On the left side, integrate \( \int \frac{dy}{\sqrt{1-y^2}} \), which is the inverse sine function, resulting in \( \sin^{-1}(y) + C_1 \). On the right side, integrate \( \int x \, dx \), which is \( \frac{x^2}{2} + C_2 \).

Step 4: Combine the constants

Combine the integration constants \( C_1 \) and \( C_2 \) into a single constant \( C \), leading to the equation \( \sin^{-1}(y) = \frac{x^2}{2} + C \).

Step 5: Solve for y

Solve for \( y \) by applying the sine function to both sides of the equation: \( y = \sin \left( \frac{x^2}{2} + C \right) \).

Step 6: Conclusion: General solution

The general solution of the differential equation is \( y = \sin \left( \frac{x^2}{2} + C \right) \), where \( C \) is the constant of integration.

Key Concepts

Separation of Variables

The method of separation of variables is an effective technique for solving differential equations. It works by rearranging the equation to separate the differing variables, in our case,

• one side of the equation will consist entirely of terms involving only the dependent variable, like \( y \),
• while the other side will consist of terms involving the independent variable, like \( x \).

This step is crucial because, after separation, each side can be independently integrated. In the provided exercise, we started with the equation \( \frac{dy}{dx} = x \sqrt{1 - y^2} \). We then rearranged it to isolate the terms involving \( y \) on one side, resulting in \( \frac{dy}{\sqrt{1-y^2}} = x \, dx \).
This process ensures that we can now move to the integration phase more easily.

Integration Techniques

Integration is the next step after separating the variables in a differential equation. Depending on the structure of the separated terms, we employ different techniques:

• The left side of the equation, \( \int \frac{dy}{\sqrt{1-y^2}} \), involves a function of \( y \). This is an inverse trigonometric function, recognized as the derivative of \( \sin^{-1}(y) \), leading to the result \( \sin^{-1}(y) + C_1 \).
• The right side, \( \int x \, dx \), is a simpler polynomial integration, resulting in \( \frac{x^2}{2} + C_2 \).

Combining the two integration results leads us to connect both results with equality: \( \sin^{-1}(y) = \frac{x^2}{2} + C \), where \( C \) represents a constant.

General Solution

After integrating both sides, the goal is to derive the general solution which describes \( y \) in terms of \( x \).

• The equation \( \sin^{-1}(y) = \frac{x^2}{2} + C \) describes the relation between \( y \) and \( x \) after integration.
• Solving this for \( y \) involves applying the sine function to isolate \( y \), resulting in \( y = \sin \left( \frac{x^2}{2} + C \right) \).

This \( y \) expression in terms of \( x \) represents the general solution to the original differential equation, showcasing all possible solutions depending on the constant \( C \). Understanding general solutions is vital as they provide a family of curves that can be narrowed down to a specific solution with given initial conditions.