L?sen Sie die folgenden Differentialgleichungen 1 . Ordnung (gemischte Aufgaben): a) \(y^{\prime}=x\left(y^{2}+1\right)\) b) \(y^{\prime}=y \cdot \sin x\) c) \(y^{\prime}=x y\) d) \(x y^{\prime}+y=2 \cdot \ln x\) e) \(y^{\prime}=5 x^{4}(y+1)\) f) \(y^{\prime}-5 y=2 \cdot \cos x-\sin (3 x)\)

First order differential equations are equations that involve the first derivative of a function but no higher derivatives. They can be represented in the form:

• \( \frac{dy}{dx} = f(x, y) \)

In simpler terms, they describe relationships where the rate of change of a function depends only on the function itself and the independent variable.

Some common types of first order differential equations include:

• Separable equations
• Linear equations
• Exact equations
• Homogeneous equations
• Bernoulli equations

First order differential equations are fundamental in modeling various natural and physical processes, such as population growth, radioactive decay, and cooling/heating laws. Mastering first-order differential equations is essential because they form the building block for more complex differential equations.

The technique of separation of variables is a powerful method for solving first order differential equations. It involves rearranging the equation so that each variable appears on a different side of the equation. The goal is to express the differential equation in the form:

• \( \frac{dy}{dx} = g(x)h(y) \)
• \( \frac{1}{h(y)}dy = g(x)dx \)

Here is a step-by-step approach for solving using separation of variables:

1. Rewrite the equation: Move all terms involving one variable (y) to one side, and all terms involving the other variable (x) to the other side.
2. Integrate both sides: Integrate the equation with respect to their respective variables.
3. Add constant of integration: After integrating, add the constant of integration (C). This often appears after the integration process and varies depending on the equation.
4. Solve for y: If possible, solve the resulting expression for the dependent variable y.

This method is applicable to many different forms of equations and is a critical tool for anyone working with differential equations.

Linear differential equations are a specific type of differential equation where the dependent variable and its derivatives appear linearly. The general form of a first-order linear differential equation is:

• \( y' + P(x)y = Q(x) \)

Where \( P(x) \) and \( Q(x) \) are functions of the independent variable. To solve linear differential equations, use the method of integrating factors. Here’s how it works:

1. Identify P(x): Note the coefficient function of y, denoted as \( P(x) \).
2. Compute the integrating factor \( \, \mu(x) \): This factor is given by \( \mu(x) = e^{ \, \int P(x)dx} \).
3. Multiply the differential equation by \( \, \mu(x) \): This will transform the left side of the equation into the derivative of a product.
4. Integrate both sides: The equation now transforms into \( d/dx[ \mu(x)y] = \mu(x)Q(x) \). Integrate both sides with respect to x.
5. Solve for y: Solve the resulting equation for y to find the general solution.

Linear differential equations appear in various fields like physics, engineering, and economics. Understanding them is invaluable for applications like circuit analysis, population dynamics, and heat transfer.

An integrating factor is a function used to simplify the process of solving linear first-order differential equations. The integrating factor, usually denoted as \( \,\mu(x) \), helps transform the differential equation into an exact equation, making it easier to integrate. To find the integrating factor, follow these steps:

1. Write the equation in standard form: Ensure the differential equation is in the form \( y' + P(x)y = Q(x) \).
2. Calculate \( \,\mu(x) \): Compute the integrating factor using the formula \( \,\mu(x) = e^{ \int P(x)dx} \).
3. Multiply through by \( \,\mu(x) \): Apply the integrating factor to both sides of the equation to obtain \( \mu(x)y' + \mu(x)P(x)y = \mu(x)Q(x) \).
4. Simplify the left-hand side: Recognize that the left-hand side of the equation is now \( \frac{d}{dx}[ \mu(x)y] \, \), which can be integrated directly.
5. Integrate both sides: Integrate the simplified equation with respect to x to find an expression for y.

Integrating factors are especially useful because they simplify the process of solving linear equations and are a standard tool in the toolkit of anyone working with differential equations.