What is the general solution of the differential equation \(\frac{{dy}}{{dx}} + \frac{x}{y} = 0\) ?
x 2+ y 2= c
Understanding and Solving Differential Equations
The question asks for the general solution of the given differential equation \(\frac{{dy}}{{dx}} + \frac{x}{y} = 0\). This is a first-order differential equation, and we can determine its type to apply the appropriate solution method.
Analyzing the Given Differential Equation
The given equation is:
\[\frac{{dy}}{{dx}} + \frac{x}{y} = 0\]
We can rearrange this equation to isolate the \(\frac{{dy}}{{dx}}\) term:
\[\frac{{dy}}{{dx}} = -\frac{x}{y}\]
This form shows that the derivative \(\frac{{dy}}{{dx}}\) is expressed as a function of \(x\) and \(y\). Specifically, it can be written in the form \(\frac{{dy}}{{dx}} = f(x, y)\).
Solving the Differential Equation by Separation of Variables
The rearranged equation \(\frac{{dy}}{{dx}} = -\frac{x}{y}\) is a separable differential equation because we can rewrite it such that all terms involving \(y\) are on one side with \(dy\), and all terms involving \(x\) are on the other side with \(dx\).
Multiply both sides by \(y\) and by \(dx\):
\[y \, dy = -x \, dx\]
Now that the variables are separated, we can integrate both sides of the equation:
\[\int y \, dy = \int -x \, dx\]
Performing the integration:
\[\frac{y^2}{2} = -\frac{x^2}{2} + C'\]
where \(C'\) is the constant of integration.
Finding the General Solution
To get the general solution in a standard form, we can rearrange the integrated equation. Move the term involving \(x^2\) to the left side:
\[\frac{y^2}{2} + \frac{x^2}{2} = C'\]
Multiply the entire equation by 2 to remove the denominators:
\[y^2 + x^2 = 2C'\]
Since \(C'\) is an arbitrary constant, \(2C'\) is also an arbitrary constant. Let's denote \(2C'\) as a new constant, say \(c\). So, the general solution is:
\[x^2 + y^2 = c\]
This equation represents the family of curves that are solutions to the given differential equation. These curves are circles centered at the origin with radius \(\sqrt{c}\) (for \(c > 0\)).
Comparing with Options
Let's compare our derived general solution with the given options:
Option 1: \(x^2 + y^2 = c\)
Option 2: \(x^2 - y^2 = c\)
Option 3: \(x^2 + y^2 = cxy\)
Option 4: \(x + y = c\)
Our derived solution \(x^2 + y^2 = c\) exactly matches Option 1.
| Step | Process | Equation |
|---|---|---|
| 1 | Given DE | \(\frac{{dy}}{{dx}} + \frac{x}{y} = 0\) |
| 2 | Rearrange | \(\frac{{dy}}{{dx}} = -\frac{x}{y}\) |
| 3 | Separate Variables | \(y \, dy = -x \, dx\) |
| 4 | Integrate both sides | \(\int y \, dy = \int -x \, dx\) |
| 5 | Perform Integration | \(\frac{y^2}{2} = -\frac{x^2}{2} + C'\) |
| 6 | Rearrange and Simplify | \(x^2 + y^2 = c\) |
Thus, the general solution of the differential equation \(\frac{{dy}}{{dx}} + \frac{x}{y} = 0\) is \(x^2 + y^2 = c\).
Revision Table: Differential Equations Concepts
| Concept | Description | Example Form |
|---|---|---|
| Differential Equation (DE) | An equation involving derivatives of a function. | \(\frac{dy}{dx} = f(x,y)\) |
| Order of DE | The highest order of the derivative present in the equation. | \(\frac{d^2y}{dx^2} + y = 0\) (Order 2) |
| Separable DE | A first-order DE that can be written as \(M(x)dx + N(y)dy = 0\). | \(\frac{dy}{dx} = \frac{g(x)}{h(y)}\) which becomes \(h(y)dy = g(x)dx\) |
| General Solution | A solution involving an arbitrary constant (or constants) that represents all possible solutions. | \(y = Ce^x\) for \(\frac{dy}{dx} = y\) |
Additional Information on Solving Differential Equations
Solving differential equations is a fundamental topic in calculus and applied mathematics. The method used depends heavily on the type of differential equation.
First-Order DEs: Common types include separable, linear (first-order), exact, and homogeneous equations. Each type has specific steps for finding the general solution.
Separable Equations: These are the simplest to solve. The process always involves separating the variables and then integrating both sides independently. The constant of integration is crucial for the general solution.
Initial Conditions: If an initial condition (e.g., \(y(x_0) = y_0\)) is provided, the arbitrary constant \(c\) in the general solution can be determined, yielding a particular solution.
The differential equation \(\frac{{dy}}{{dx}} + \frac{x}{y} = 0\) is a classic example of a separable equation, making the separation of variables method the most straightforward approach to find its general solution \(x^2 + y^2 = c\).
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