Solve The Differential Equation 1 4 X D Y D X = Y 2 X 2 − 1 \frac{1}{4x} \frac{dy}{dx} = Y \sqrt{2x^2-1} 4 X 1 ​ D X D Y ​ = Y 2 X 2 − 1 ​ .A. E 2 3 ( 2 X 2 − 1 ) 3 E^{\frac{2}{3 \sqrt{(2x^2-1)^3}}} E 3 ( 2 X 2 − 1 ) 3 ​ 2 ​ B. C E 2 3 ( 2 X 2 − 1 ) 3 C E^{\frac{2}{3} \sqrt{(2x^2-1)^3}} C E 3 2 ​ ( 2 X 2 − 1 ) 3 ​ C. E 2 3 ( 2 X 2 − 1 ) 3 + C E^{\frac{2}{3} \sqrt{(2x^2-1)^3}} + C E 3 2 ​ ( 2 X 2 − 1 ) 3 ​ + C D. None Of

Introduction

Differential equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific differential equation, $\frac{1}{4x} \frac{dy}{dx} = y \sqrt{2x^2-1}$, and explore the different methods and techniques used to arrive at the solution.

Understanding the Differential Equation

The given differential equation is $\frac{1}{4x} \frac{dy}{dx} = y \sqrt{2x^2-1}$. To solve this equation, we need to isolate the derivative $\frac{dy}{dx}$ and then integrate both sides to find the general solution.

Separating the Variables

The first step in solving the differential equation is to separate the variables. We can do this by multiplying both sides of the equation by $4x$ and dividing both sides by $y$. This gives us:

$$\frac{dy}{dx} = 4x \cdot y \sqrt{2x^2-1}$$

Rearranging the Equation

Next, we can rearrange the equation to isolate the derivative $\frac{dy}{dx}$. We can do this by dividing both sides of the equation by $y$ and multiplying both sides by $dx$. This gives us:

$$\frac{dy}{y} = 4x \cdot \sqrt{2x^2-1} \, dx$$

Integrating Both Sides

Now that we have separated the variables and isolated the derivative, we can integrate both sides of the equation. We can do this by using the power rule of integration and the substitution method.

$$\int \frac{dy}{y} = \int 4x \cdot \sqrt{2x^2-1} \, dx$$

Evaluating the Integrals

To evaluate the integrals, we can use the power rule of integration and the substitution method. We can start by evaluating the integral on the left-hand side:

$$\int \frac{dy}{y} = \ln |y| + C$$

Evaluating the Integral on the Right-Hand Side

Next, we can evaluate the integral on the right-hand side. We can do this by using the substitution method. Let $u = 2x^2 - 1$. Then, $du = 4x \, dx$. We can substitute these values into the integral:

$$\int 4x \cdot \sqrt{2x^2-1} \, dx = \int \sqrt{u} \, du$$

Evaluating the Integral

To evaluate the integral, we can use the power rule of integration. We can do this by raising the power of the variable $u$ by one and dividing the coefficient by the new power:

$$\int \sqrt{u} \, du = \frac{2}{3} u^{\frac{3}{2}} + C$$

Substituting Back

Now that we have evaluated the integral, we can substitute back to find the solution. We can do this by substituting $u = 2x^2 - 1$ back into the solution:

$$\frac{2}{3} u^{\frac{3}{2}} + C = \frac{2}{3} (2x^2 - 1)^{\frac{3}{2}} + C$$

Simplifying the Solution

Finally, we can simplify the solution by combining like terms:

$$\frac{2}{3} (2x^2 - 1)^{\frac{3}{2}} + C = \frac{2}{3} \sqrt{(2x^2-1)^3} + C$$

Conclusion

In this article, we have solved the differential equation $\frac{1}{4x} \frac{dy}{dx} = y \sqrt{2x^2-1}$ using the power rule of integration and the substitution method. We have shown that the solution is $\frac{2}{3} \sqrt{(2x^2-1)^3} + C$. This solution is a general solution, and it satisfies the original differential equation.

Answer

The correct answer is C. $e^{\frac{2}{3} \sqrt{(2x^2-1)^3}} + C$

Introduction

In our previous article, we solved the differential equation $\frac{1}{4x} \frac{dy}{dx} = y \sqrt{2x^2-1}$ using the power rule of integration and the substitution method. In this article, we will provide a Q&A approach to help you understand the solution and the steps involved in solving the differential equation.

Q: What is the first step in solving the differential equation?

A: The first step in solving the differential equation is to separate the variables. We can do this by multiplying both sides of the equation by $4x$ and dividing both sides by $y$. This gives us:

$$\frac{dy}{dx} = 4x \cdot y \sqrt{2x^2-1}$$

Q: How do we isolate the derivative $\frac{dy}{dx}$?

A: We can isolate the derivative $\frac{dy}{dx}$ by dividing both sides of the equation by $y$ and multiplying both sides by $dx$. This gives us:

$$\frac{dy}{y} = 4x \cdot \sqrt{2x^2-1} \, dx$$

Q: What is the next step in solving the differential equation?

A: The next step in solving the differential equation is to integrate both sides of the equation. We can do this by using the power rule of integration and the substitution method.

Q: How do we evaluate the integral on the left-hand side?

A: We can evaluate the integral on the left-hand side by using the power rule of integration. We can do this by raising the power of the variable $y$ by one and dividing the coefficient by the new power:

$$\int \frac{dy}{y} = \ln |y| + C$$

Q: How do we evaluate the integral on the right-hand side?

A: We can evaluate the integral on the right-hand side by using the substitution method. Let $u = 2x^2 - 1$. Then, $du = 4x \, dx$. We can substitute these values into the integral:

$$\int 4x \cdot \sqrt{2x^2-1} \, dx = \int \sqrt{u} \, du$$

Q: How do we evaluate the integral?

A: We can evaluate the integral by using the power rule of integration. We can do this by raising the power of the variable $u$ by one and dividing the coefficient by the new power:

$$\int \sqrt{u} \, du = \frac{2}{3} u^{\frac{3}{2}} + C$$

Q: What is the final solution to the differential equation?

A: The final solution to the differential equation is $\frac{2}{3} \sqrt{(2x^2-1)^3} + C$.

Q: What is the significance of the constant $C$?

A: The constant $C$ is a family of solutions to the differential equation. It represents all possible solutions to the differential equation, and it is used to account for the arbitrary constant that arises when integrating the differential equation.

Conclusion

In this article, we have provided a Q&A approach to help you understand the solution and the steps involved in solving the differential equation $\frac{1}{4x} \frac{dy}{dx} = y \sqrt{2x^2-1}$. We have shown that the solution is $\frac{2}{3} \sqrt{(2x^2-1)^3} + C$, and we have explained the significance of the constant $C$.

Frequently Asked Questions

• Q: What is the first step in solving the differential equation? A: The first step in solving the differential equation is to separate the variables.
• Q: How do we isolate the derivative $\frac{dy}{dx}$? A: We can isolate the derivative $\frac{dy}{dx}$ by dividing both sides of the equation by $y$ and multiplying both sides by $dx$.
• Q: What is the next step in solving the differential equation? A: The next step in solving the differential equation is to integrate both sides of the equation.
• Q: How do we evaluate the integral on the left-hand side? A: We can evaluate the integral on the left-hand side by using the power rule of integration.
• Q: How do we evaluate the integral on the right-hand side? A: We can evaluate the integral on the right-hand side by using the substitution method.

Additional Resources

• Differential Equations: A First Course
• Calculus: Early Transcendentals
• Mathematics: A Very Short Introduction

Conclusion

In this article, we have provided a Q&A approach to help you understand the solution and the steps involved in solving the differential equation $\frac{1}{4x} \frac{dy}{dx} = y \sqrt{2x^2-1}$. We have shown that the solution is $\frac{2}{3} \sqrt{(2x^2-1)^3} + C$, and we have explained the significance of the constant $C$. We hope that this article has been helpful in understanding the solution and the steps involved in solving the differential equation.