The solution of the differential equation dy = (1 + y 2) dx is
y = tan (x + c)
Solving the Differential Equation \(dy = (1 + y^2) dx\)
We are asked to find the solution to the differential equation given by \(dy = (1 + y^2) dx\). This is a first-order differential equation.
The given differential equation can be rewritten as:
\(\frac{dy}{dx} = 1 + y^2\)
This is a separable differential equation because we can separate the variables \(y\) and \(x\) on opposite sides of the equation.
Applying Separation of Variables
To separate the variables, we can move the term involving \(y\) to the left side and the term involving \(x\) (which is just \(dx\)) to the right side:
\(\frac{dy}{1 + y^2} = dx\)
Integrating Both Sides
Now, we integrate both sides of the separated equation:
\(\int \frac{dy}{1 + y^2} = \int dx\)
The integral of \(\frac{1}{1 + y^2}\) with respect to \(y\) is a standard integral, which is \(\tan^{-1}(y)\) or \(\arctan(y)\). The integral of \(1\) with respect to \(x\) is \(x\). Remember to add a constant of integration, let's call it \(c\), to one side (usually the side with the independent variable, \(x\)).
So, integrating both sides gives:
\(\tan^{-1}(y) = x + c\)
Solving for \(y\)
The solution is currently in the form \(\tan^{-1}(y) = x + c\). To express \(y\) explicitly in terms of \(x\), we take the tangent of both sides of the equation:
\(\tan(\tan^{-1}(y)) = \tan(x + c)\)
This simplifies to:
\(y = \tan(x + c)\)
This is the general solution to the given differential equation \(dy = (1 + y^2) dx\).
Comparing with Options
Let's compare our derived solution \(y = \tan(x + c)\) with the provided options:
- \(y = \tan x + c\) - This is different because the constant \(c\) is outside the tangent function.
- \(y = \tan (x + c)\) - This matches our derived solution exactly.
- \(\tan^{-1} (y + c) = x\) - This is different from our intermediate step \(\tan^{-1}(y) = x + c\).
- \(\tan^{-1} (y + c) = 2x\) - This is also different from our intermediate step.
Therefore, the correct solution is \(y = \tan (x + c)\).
Revision Table: Key Steps in Solving Separable Differential Equations
| Step | Description | Application to \(dy = (1 + y^2) dx\) |
|---|---|---|
| 1 | Rewrite the equation as \(\frac{dy}{dx} = f(x)g(y)\). | \(\frac{dy}{dx} = 1 \cdot (1 + y^2)\) |
| 2 | Separate variables: \(\frac{dy}{g(y)} = f(x) dx\). | \(\frac{dy}{1 + y^2} = dx\) |
| 3 | Integrate both sides: \(\int \frac{dy}{g(y)} = \int f(x) dx\). | \(\int \frac{dy}{1 + y^2} = \int dx\) |
| 4 | Solve the integrals and add a constant of integration \(c\). | \(\tan^{-1}(y) = x + c\) |
| 5 | Solve for \(y\) (if possible) to get the general solution. | \(y = \tan(x + c)\) |
Additional Information: Understanding Differential Equations and Solutions
A differential equation is an equation that relates a function with one or more of its derivatives. The order of a differential equation is the order of the highest derivative appearing in it. The given equation \(dy/dx = 1 + y^2\) is a first-order ordinary differential equation.
A solution to a differential equation is a function that satisfies the equation when substituted into it. A general solution contains arbitrary constants (like \(c\) in our case), which arise from the integration process. A particular solution is obtained from the general solution by assigning specific values to the constants, usually based on initial or boundary conditions.
Separation of variables is a common technique used to solve first-order differential equations that can be written in the form \(\frac{dy}{dx} = f(x)g(y)\). This method works because the terms involving \(y\) can be moved to one side with \(dy\), and the terms involving \(x\) can be moved to the other side with \(dx\), allowing for independent integration.
The function \(\tan^{-1}(y)\) (or arctan y) is the inverse tangent function. Its derivative with respect to \(y\) is \(\frac{1}{1+y^2}\). This is why integrating \(\frac{1}{1+y^2}\) gives \(\tan^{-1}(y)\).
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