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Magazine Chapter 2: Problem 8
$$ \left(2 x y+x^{2}+x^{4}\right) d x-\left(1+x^{2}\right) d y=0 $$
Short Answer
Expert verified
The implicit solution is \( xy^2 + \frac{x^3}{3} + \frac{x^5}{5} + y = C \).
Step by step solution
01
Rewrite the Given Equation
The given differential equation is \( \left(2xy + x^2 + x^4\right) dx - \left(1 + x^2\right) dy = 0 \). First, identify the functions for \( M(x, y) = 2xy + x^2 + x^4 \) and \( N(x, y) = 1 + x^2 \).
02
Check for Exactness
To check if the equation is exact, take the partial derivative of \( M \) with respect to \( y \) and \( N \) with respect to \( x \). We have \( \frac{\partial M}{\partial y} = 2x \) and \( \frac{\partial N}{\partial x} = 2x \). Since these are equal, the differential equation is exact.
03
Integrate M with Respect to x
Integrate \( M(x, y) = 2xy + x^2 + x^4 \) with respect to \( x \): \[ \int (2xy + x^2 + x^4) \, dx = xy^2 + \frac{x^3}{3} + \frac{x^5}{5} + h(y). \] Here, \( h(y) \) is a function of \( y \) alone.
04
Determine h(y) by Comparing with N
Now differentiate \( xy^2 + \frac{x^3}{3} + \frac{x^5}{5} + h(y) \) with respect to \( y \): \( x + h'(y) \). Set this equal to \( N(x, y) = 1 + x^2 \), thus: \( h'(y) = 1 \). Integrate to find \( h(y) = y + C \), where \( C \) is a constant.
05
Form the Solution
Substitute \( h(y) = y \) back into the integral result: \[ xy^2 + \frac{x^3}{3} + \frac{x^5}{5} + y = C. \] This is the implicit solution to the differential equation.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Partial Derivatives
Partial derivatives are a fundamental tool in calculus used to find the rate at which a function changes with respect to one of its variables while keeping the other variables constant. In our differential equation, we have functions defined as \( M(x, y) = 2xy + x^2 + x^4 \) and \( N(x, y) = 1 + x^2 \). To check for exactness, we use partial derivatives.
- For \( M \), we take the partial derivative with respect to \( y \), denoted as \( \frac{\partial M}{\partial y} \).
- For \( N \), we take the partial derivative with respect to \( x \), denoted as \( \frac{\partial N}{\partial x} \).
Integration
Integration is the process of finding the integral of a function, which is essentially the reverse operation of taking a derivative. In an exact differential equation, once you've verified exactness through partial derivatives, the next step is integration.Our task is to integrate \( M(x, y) = 2xy + x^2 + x^4 \) with respect to \( x \):\[ \int (2xy + x^2 + x^4) \, dx = xy^2 + \frac{x^3}{3} + \frac{x^5}{5} + h(y). \]
- Each term of the function is integrated individually.
- The \( h(y) \) term is added post-integration, representing an "arbitrary constant" that can depend on \( y \).
Implicit Solutions
An implicit solution is one where the relationship between \( x \) and \( y \) is expressed in a single equation, as opposed to an explicit function \( y = f(x) \). Implicit solutions often arise in exact differential equations.After integrating and finding \( h(y) \), the solution appears:\[ xy^2 + \frac{x^3}{3} + \frac{x^5}{5} + y = C, \]where \( C \) is a constant determined by initial conditions or boundary values.
- The solution expresses \( x \) and \( y \) as related through a single, simple equation.
- It is part of a larger family of solutions where \( C \) can vary.
Exactness Condition
The exactness condition is a mathematical criterion used to determine if a differential equation can be solved using the methods for exact equations. It ensures that the equation is suitable for integration, leading to a direct path to finding the solution.In our original problem, the condition is based on equal partial derivatives: \( \frac{\partial M}{\partial y} = \frac{\partial N}{\partial x} \).
- This analysis leads to the determination that the equation is exact if these partial derivatives equal each other.
- Exactness implies there exists a function \( \Psi(x, y) \) such that \( d\Psi = M dx + N dy = 0 \).
