What is the solution of the differential equation (dy − dx) + cos x(dy + dx) = 0 ?
Solving Differential Equations: Step-by-Step Guide
The given problem asks for the solution to the differential equation \( (dy − dx) + \cos x(dy + dx) = 0 \). This is a first-order differential equation that can be solved by rearranging the terms and integrating.
Rearranging the Differential Equation
First, let's expand the equation and group the terms involving \(dy\) and \(dx\):
The given equation is:
\( dy - dx + \cos x \cdot dy + \cos x \cdot dx = 0 \)
Group the \(dy\) terms and the \(dx\) terms:
\( (1 + \cos x) dy + (\cos x - 1) dx = 0 \)
Move the \(dx\) term to the other side:
\( (1 + \cos x) dy = - (\cos x - 1) dx \)
\( (1 + \cos x) dy = (1 - \cos x) dx \)
Separating Variables
Now, we can separate the variables \(y\) and \(x\). Divide both sides by \( (1 + \cos x) \) and by \( dx \) (implicitly, by writing \( \frac{dy}{dx} \)), but it's easier to just separate \(dy\) and \(dx\):
\( dy = \frac{1 - \cos x}{1 + \cos x} dx \)
This is a separable differential equation where the variables \(y\) and \(x\) are on opposite sides of the equation.
Using Trigonometric Identities
To integrate the right side, we can use the half-angle trigonometric identities:
- \( 1 - \cos x = 2 \sin^2 \left(\frac{x}{2}\right) \)
- \( 1 + \cos x = 2 \cos^2 \left(\frac{x}{2}\right) \)
Substitute these identities into the expression:
\( \frac{1 - \cos x}{1 + \cos x} = \frac{2 \sin^2 \left(\frac{x}{2}\right)}{2 \cos^2 \left(\frac{x}{2}\right)} = \tan^2 \left(\frac{x}{2}\right) \)
So, the differential equation becomes:
\( dy = \tan^2 \left(\frac{x}{2}\right) dx \)
Now, use the identity \( \tan^2 \theta = \sec^2 \theta - 1 \):
\( dy = \left(\sec^2 \left(\frac{x}{2}\right) - 1\right) dx \)
Integrating Both Sides
Integrate both sides of the equation:
\( \int dy = \int \left(\sec^2 \left(\frac{x}{2}\right) - 1\right) dx \)
The left side integral is straightforward:
\( \int dy = y + C_1 \)
The right side integral can be split into two parts:
\( \int \left(\sec^2 \left(\frac{x}{2}\right) - 1\right) dx = \int \sec^2 \left(\frac{x}{2}\right) dx - \int 1 \, dx \)
For the first integral, \( \int \sec^2 \left(\frac{x}{2}\right) dx \), let \(u = \frac{x}{2}\). Then \(du = \frac{1}{2} dx\), which means \(dx = 2 du\).
So, \( \int \sec^2 \left(\frac{x}{2}\right) dx = \int \sec^2(u) (2 du) = 2 \int \sec^2 u \, du = 2 \tan u + C_2 \). Substitute back \(u = \frac{x}{2}\):
\( 2 \tan \left(\frac{x}{2}\right) + C_2 \)
The second integral is:
\( \int 1 \, dx = x + C_3 \)
Combining the results for the right side integral:
\( \int \left(\sec^2 \left(\frac{x}{2}\right) - 1\right) dx = 2 \tan \left(\frac{x}{2}\right) - x + C_2 - C_3 \)
Now, equate the integrals of both sides:
\( y + C_1 = 2 \tan \left(\frac{x}{2}\right) - x + C_2 - C_3 \)
Combine the constants into a single constant \(C = C_2 - C_3 - C_1\):
\( y = 2 \tan \left(\frac{x}{2}\right) - x + C \)
Verification with Options
Let's compare our solution with the given options:
| Option | Expression |
| 1 | \(y = \tan \left(\frac{x}{2}\right) - x + c\) |
| 2 | \(y = \frac{1}{2}\tan\left(\frac{x}{2}\right) - x + c\) |
| 3 | \(y = 2\tan \left(\frac{x}{2}\right) - x + c\) |
| 4 | \(y = \tan \left(\frac{x}{2}\right) - 2x + c\) |
Our derived solution is \(y = 2 \tan \left(\frac{x}{2}\right) - x + C\), which matches Option 3.
Revision Table: Key Concepts in Solving Differential Equations
| Concept | Description | Relevance to Problem |
| Differential Equation | An equation involving an unknown function and its derivatives. | The given equation is a first-order ordinary differential equation. |
| Separable Equation | A first-order differential equation that can be written in the form \( \frac{dy}{dx} = f(x)g(y) \) or \( M(x) dx + N(y) dy = 0 \). | The given equation was transformed into the separable form \( dy = \tan^2\left(\frac{x}{2}\right) dx \). |
| Integration | The process of finding the antiderivative of a function; used to solve separable differential equations. | We integrated both sides of the separated equation to find the solution \(y(x)\). |
| Trigonometric Identities | Equations involving trigonometric functions that are true for all values of the variables. | Identities for \(1 \pm \cos x\) and \( \tan^2 \theta \) were crucial for simplifying the integrand. |
| Constant of Integration | An arbitrary constant introduced when performing indefinite integration. | Represented by \(C\) in the final solution, accounting for the family of solutions. |
Additional Information: Solving Differential Equations
Differential equations are fundamental in many fields of science and engineering, describing processes involving rates of change. The method used here, separation of variables, is applicable when the equation can be written with all terms involving \(y\) and \(dy\) on one side and all terms involving \(x\) and \(dx\) on the other.
Key steps often include:
- Rearranging the equation into a standard form.
- Identifying the type of differential equation (e.g., separable, linear, exact).
- Applying the appropriate solution method.
- Integrating both sides (for separable equations).
- Solving for the dependent variable (usually \(y\)).
- Including the constant of integration.
In this specific problem, recognizing the trigonometric identities for \(1 \pm \cos x\) in terms of half angles was key to simplifying the integrand \( \frac{1 - \cos x}{1 + \cos x} \) into a form (\( \tan^2 \left(\frac{x}{2}\right) \)) that could be easily integrated after applying the identity \( \tan^2 \theta = \sec^2 \theta - 1 \).
Understanding these techniques is vital for solving a wide range of problems involving differential equations.
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