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Magazine Chapter 2: Problem 2
Solve the following as linear equations: \&. \(x \frac{d y}{d x}-3 y=x^{4}\); ?. \(y^{\prime}+y=2 x e^{-x}+x^{2}\) b. \(y^{\prime}+y=\frac{1}{1+e^{2 x}} ;\) Q. \(y^{\prime}+y \cot x=2 x \csc x\) \(\&\left(1+x^{2}\right) d y+2 x y d x=\cot x d x\) f. \(\left(2 y-x^{3}\right) d x=x d y\)
Short Answer
Expert verified
\( y = e^{-x} \arctan(e^x) + Ce^{-x} \).
Step by step solution
01
Identify the Type of Differential Equation
The equation given is \( y' + y = \frac{1}{1+e^{2x}} \). This is a first-order linear ordinary differential equation, which has the standard form \( y' + p(x)y = q(x) \). In our case, \( p(x) = 1 \) and \( q(x) = \frac{1}{1+e^{2x}} \).
02
Find the Integrating Factor
To solve the equation, we find the integrating factor \( \mu(x) \), which is given by \( e^{\int p(x) dx} \). Here, \( p(x) = 1 \), so the integrating factor is \( \mu(x) = e^{\int 1 \cdot dx} = e^x \).
03
Multiply Through by the Integrating Factor
Multiply every term in the differential equation by the integrating factor \( e^x \). This gives: \[ e^x y' + e^x y = e^x \cdot \frac{1}{1 + e^{2x}} \]
04
Rewrite the Left Side as a Derivative
The left side of the equation becomes the derivative of \( e^x y \) with respect to \( x \). Rewrite the equation: \[ \frac{d}{dx}(e^x y) = \frac{e^x}{1 + e^{2x}} \]
05
Integrate Both Sides
Integrate both sides with respect to \( x \). This gives: \[ e^x y = \int \frac{e^x}{1 + e^{2x}} \, dx + C \]
06
Solve the Integral
The integral \( \int \frac{e^x}{1 + e^{2x}} \, dx \) requires substitution or partial fraction techniques. Let \( u = e^x \), then \( du = e^x dx \), and this transforms the integral to \( \int \frac{1}{1 + u^2} \, du \), which is \( \arctan(u) + C \). So, \( \int \frac{e^x}{1 + e^{2x}} \, dx = \arctan(e^x) + C \).
07
Substitute Back into Equation
Substitute back into the equation: \[ e^x y = \arctan(e^x) + C \]
08
Solve for y
Divide both sides by \( e^x \) to solve for \( y \): \[ y = e^{-x} \arctan(e^x) + Ce^{-x} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Ordinary Differential Equations
Ordinary differential equations (ODEs) are equations involving a function and its derivatives. They are called "ordinary" to distinguish them from partial differential equations, which involve multiple independent variables. In ODEs, the derivatives are with respect to a single variable, often time or space. These equations play a crucial role in modeling natural phenomena, engineering systems, and various scientific processes.
Common applications include:
Common applications include:
- Population dynamics in ecology.
- Electrical circuits in engineering.
- Motion of objects in physics.
First-order Differential Equations
First-order differential equations involve the first derivative of a function but not any higher derivatives. These are among the simplest types of differential equations, providing a starting point to understand more complex systems. The general form is typically written as \( y' + p(x)y = q(x) \), where \( y' \) represents the first derivative of \( y \) with respect to \( x \).
In these equations:
In these equations:
- \( y' \) represents the rate of change of \( y \).
- \( p(x) \) is a function of the independent variable \( x \), representing the coefficient of \( y \).
- \( q(x) \) is a source term that depends on \( x \).
Integrating Factor
The integrating factor is a technique used to solve first-order linear differential equations. It simplifies these equations by transforming them into a form that can easily be integrated. The integrating factor, typically denoted as \( \mu(x) \), is chosen to make the left-hand side of the differential equation a product derivative.
To determine the integrating factor:
To determine the integrating factor:
- Calculate \( \mu(x) = e^{\int p(x) \, dx} \).
- Multiply every term of the differential equation by \( \mu(x) \).
- Rewrite the left side as a derivative of a product.
Solving Differential Equations
Solving differential equations involves finding a function that satisfies the equation under consideration. The process can vary depending on whether the differential equation is linear or nonlinear, homogeneous or non-homogeneous, and the order of the equation.
General steps for solving first-order linear equations using an integrating factor include:
General steps for solving first-order linear equations using an integrating factor include:
- Identify the equation's form and check if it's linear.
- Find the integrating factor \( \mu(x) \).
- Transform the equation with the integrating factor, converting it to a solvable format.
- Integrate both sides with respect to the independent variable.
- Solve for the original function.
