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 Ce^{\frac{2}{3 \sqrt{(2x^2 - 1)^3}}} C E 3 ( 2 X 2 − 1 ) 3 ​ 2 ​ C. E 2 3 ( 2 X 2 − 1 ) 3 + C E^{\frac{2}{3 \sqrt{(2x^2 - 1)^3}}} + C E 3 ( 2 X 2 − 1 ) 3 ​ 2 ​ + C D. None Of

Introduction

Differential equations are a fundamental concept in mathematics, and solving them is a crucial aspect of various fields, including physics, engineering, and economics. 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 then dividing both sides by $y$. This gives us:

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

Integrating the Equation

Now that we have separated the variables, we can integrate both sides of the equation to find the general solution. We can start by integrating the right-hand side of the equation:

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

Using Substitution

To integrate the right-hand side of the equation, we can use substitution. Let's set $u = 2x^2 - 1$. Then, we have:

$$du = 4x dx$$

Substituting into the Integral

Now that we have substituted $u$ into the integral, we can rewrite the equation as:

$$\int \frac{dy}{y} = \int \sqrt{u} du$$

Evaluating the Integral

We can now evaluate the integral on the right-hand side:

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

Substituting Back

Now that we have evaluated the integral, we can substitute back $u = 2x^2 - 1$:

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

Finding the General Solution

Now that we have integrated both sides of the equation, we can find the general solution. We can do this by exponentiating both sides of the equation:

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

Simplifying the Solution

We can simplify the solution by combining the exponentials:

$$y = e^{\frac{2}{3} (2x^2 - 1)^{\frac{3}{2}}} e^C$$

Finding the Particular Solution

To find the particular solution, we need to determine the value of the constant $C$. We can do this by using the initial condition $y(1) = 1$.

Applying the Initial Condition

We can apply the initial condition by substituting $x = 1$ and $y = 1$ into the general solution:

$$1 = e^{\frac{2}{3} (2(1)^2 - 1)^{\frac{3}{2}}} e^C$$

Solving for C

We can solve for $C$ by rearranging the equation:

$$e^C = \frac{1}{e^{\frac{2}{3} (2(1)^2 - 1)^{\frac{3}{2}}}}$$

Finding the Particular Solution

Now that we have found the value of $C$, we can find the particular solution:

$$y = e^{\frac{2}{3} (2x^2 - 1)^{\frac{3}{2}}} \frac{1}{e^{\frac{2}{3} (2(1)^2 - 1)^{\frac{3}{2}}}}$$

Simplifying the Particular Solution

We can simplify the particular solution by combining the exponentials:

$$y = e^{\frac{2}{3} (2x^2 - 1)^{\frac{3}{2}}} \frac{1}{e^{\frac{2}{3} (1)^{\frac{3}{2}}}}$$

Final Answer

The final answer is $\boxed{Ce^{\frac{2}{3 \sqrt{(2x^2 - 1)^3}}}}$.

Introduction

Differential equations are a fundamental concept in mathematics, and solving them is a crucial aspect of various fields, including physics, engineering, and economics. 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 then dividing both sides by $y$. This gives us:

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

Integrating the Equation

Now that we have separated the variables, we can integrate both sides of the equation to find the general solution. We can start by integrating the right-hand side of the equation:

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

Using Substitution

To integrate the right-hand side of the equation, we can use substitution. Let's set $u = 2x^2 - 1$. Then, we have:

$$du = 4x dx$$

Substituting into the Integral

Now that we have substituted $u$ into the integral, we can rewrite the equation as:

$$\int \frac{dy}{y} = \int \sqrt{u} du$$

Evaluating the Integral

We can now evaluate the integral on the right-hand side:

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

Substituting Back

Now that we have evaluated the integral, we can substitute back $u = 2x^2 - 1$:

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

Finding the General Solution

Now that we have integrated both sides of the equation, we can find the general solution. We can do this by exponentiating both sides of the equation:

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

Simplifying the Solution

We can simplify the solution by combining the exponentials:

$$y = e^{\frac{2}{3} (2x^2 - 1)^{\frac{3}{2}}} e^C$$

Finding the Particular Solution

To find the particular solution, we need to determine the value of the constant $C$. We can do this by using the initial condition $y(1) = 1$.

Applying the Initial Condition

We can apply the initial condition by substituting $x = 1$ and $y = 1$ into the general solution:

$$1 = e^{\frac{2}{3} (2(1)^2 - 1)^{\frac{3}{2}}} e^C$$

Solving for C

We can solve for $C$ by rearranging the equation:

$$e^C = \frac{1}{e^{\frac{2}{3} (2(1)^2 - 1)^{\frac{3}{2}}}}$$

Finding the Particular Solution

Now that we have found the value of $C$, we can find the particular solution:

$$y = e^{\frac{2}{3} (2x^2 - 1)^{\frac{3}{2}}} \frac{1}{e^{\frac{2}{3} (2(1)^2 - 1)^{\frac{3}{2}}}}$$

Simplifying the Particular Solution

We can simplify the particular solution by combining the exponentials:

$$y = e^{\frac{2}{3} (2x^2 - 1)^{\frac{3}{2}}} \frac{1}{e^{\frac{2}{3} (1)^{\frac{3}{2}}}}$$

Final Answer

The final answer is $\boxed{Ce^{\frac{2}{3 \sqrt{(2x^2 - 1)^3}}}}$.

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Q&A

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 then dividing both sides by $y$.

Q: What is the purpose of using substitution in the integral?

A: The purpose of using substitution in the integral is to simplify the equation and make it easier to evaluate. By setting $u = 2x^2 - 1$, we can rewrite the integral as $\int \sqrt{u} du$.

Q: How do we find the particular solution?

A: To find the particular solution, we need to determine the value of the constant $C$. We can do this by using the initial condition $y(1) = 1$.

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

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

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

A: The constant $C$ is a free parameter that represents the family of solutions to the differential equation. It is used to describe the particular solution that satisfies the initial condition.

Q: How do we simplify the particular solution?

A: We can simplify the particular solution by combining the exponentials. This gives us the final answer to the differential equation.

Q: What is the relationship between the general solution and the particular solution?

A: The general solution is a family of solutions that satisfy the differential equation, while the particular solution is a specific solution that satisfies the initial condition. The particular solution is a member of the family of solutions described by the general solution.

Q: What is the purpose of using the initial condition to find the particular solution?

A: The purpose of using the initial condition is to determine the value of the constant $C$ and find the particular solution that satisfies the initial condition. This allows us to describe the specific solution that satisfies the given conditions.