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Chapter 5: Problem 14

Lösen Sie die folgenden Anfangswertaufgaben durch Variation der Konstanten: a) \(x y^{\prime}-y=x^{2} \cdot \cos x, \quad y(\pi)=2 \pi\) b) \(y^{\prime}+(\tan x) \cdot y=5 \cdot \sin (2 x), \quad\) Lösungskurve durch Punkt \(P=(3 \pi ; 2)\) c) \(x y^{\prime}+y=\ln x, \quad y(1)=1\)

Short Answer

Expert verified
Find the general solution by solving the homogeneous equation and then apply variation of parameters to find the particular solution, followed by applying the initial/boundary condition to find the specific solution.

Step by step solution

01

- Solve the Homogeneous Equation

First, solve the corresponding homogeneous differential equation by setting the non-homogeneous part to zero.
02

- Find the Particular Solution using Variation of Parameters

Assume a particular solution of the form of the solution to the homogeneous equation but with a varying constant. Substitute this into the original equation and solve for the constant.
03

- Apply the Initial/Boundary Condition

Use the given initial or boundary condition to solve for any remaining constants in the solution.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Anfangswertaufgaben
Anfangswertaufgaben, also **initial value problems**, occur when you are given a differential equation along with an initial condition. The initial condition specifies the value of the solution at a particular point.
For example, in the equation: y(π) = 2π for the equation x y' - y = x² ⋅ cos(x)
This initial condition tells us that when x = π, the value of y should be 2π. The main steps to solve an initial value problem are:
  • Solve the homogeneous differential equation first.
  • Find the particular solution that satisfies the non-homogeneous part.
  • Use the initial condition to determine any constants involved.
These concepts ensure that the solution is both mathematically coherent and adheres to the given conditions.
homogene Differentialgleichung
A homogeneous differential equation is one where the right-hand side of the equation is zero. It has the general form:
y' + p(x)y = 0,
In this form, any solution will be a family of functions of the form:
y_h = C ⋅ e^(∫ p(x) dx)
An example would be solving the homogeneous part of:
x y' - y = 0, which simplifies to:
y' = y/x. The general solution is:
y_h = Cx.
Here,
Partikuläre Lösung
The particular solution is essential for solving non-homogeneous differential equations. While the homogeneous solution deals with the complementary function, the particular solution must satisfy the non-zero part of the equation. To find it using the Variation of Parameters method:
Assume a solution similar in form to the homogeneous one but with a varying constant. For instance:
y_p = v(x) ⋅ e^(∫ p(x) dx),
where v(x) is a function to be determined.
  • Differentiate y_p and substitute it into the original equation.
  • Solve for v(x).
This particular solution is then combined with the homogeneous solution to find the general solution.
Differentialgleichungen
Differentialgleichungen, or differential equations, are equations that involve the derivatives of an unknown function. They are essential in modeling various physical, biological, and economic processes. There are two main types:
  • Ordinary Differential Equations (ODEs)
    • This type deals with functions of a single variable and their derivatives. For instance:
    dy/dx + P(x)y = Q(x),
    where dy/dx is the derivative of y with respect to x.
  • Partial Differential Equations (PDEs)
    • In contrast, these equations involve partial derivatives and functions of multiple variables. For example:
    ∂u/∂t = α∂²u/∂x².
    In summary, mastering differential equations is crucial as they provide insights into the dynamic behavior of complex systems.

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Most popular questions from this chapter

L?sen Sie die folgenden inhomogenen linearen Differentialgleichungssysteme 2. Ordnung durch , Aufsuchen einer partikulären Lösung": a) \(y_{1}^{\prime}=2 y_{2}+8 x\) $$ y_{2}^{\prime}=-2 y_{1} $$ $$ \text { b) } y_{1}^{\prime}=-y_{1}+y_{2}+4 \cdot \mathrm{e}^{2 x} $$ $$ y_{2}^{\prime}=-4 y_{1}+3 y_{2} $$

Lösen Sie die folgenden Schwingungsgleichungen (freie ungedämpfte Schwingungen): a) \(\ddot{x}+4 x=0, \quad x(0)=2, \quad \dot{x}(0)=1\) b) \(\ddot{x}+x=0, \quad x(0)=1, \quad \dot{x}(0)=-2\) c) \(\quad \ddot{x}+a^{2} x=0, \quad x(0)=0, \quad \dot{x}(0)=v_{0} \quad(a \neq 0)\)

Bestimmen Sie die allgemeinen L?sungen der folgenden inhomogenen linearen Differentialgleichungen 2. Ordnung: a) \(y^{\prime \prime}+2 y^{\prime}-3 y=3 x^{2}-4 x\) b) \(y^{\prime \prime}-y=x^{3}-2 x^{2}-4\) c) \(\ddot{x}-2 \dot{x}+x=\mathrm{e}^{2 t}\) d) \(y^{\prime \prime}-2 y^{\prime}-3 y=-2 \cdot \mathrm{e}^{3 x}\) e) \(\quad \bar{x}+10 \dot{x}+25 x=3 \cdot \cos (5 t)\) f) \(y^{\prime \prime}+10 y^{\prime}-24 y=2 x^{2}-6 x\) g) \(\quad \vec{x}-x=t \cdot \sin t\) h) \(y^{\prime \prime}+12 y^{\prime}+36 y=3 \cdot \mathrm{e}^{-6 x}\) i) \(y^{\prime \prime}+4 y=10 \cdot \sin (2 x)+2 x^{2}-x+e^{-x}\) j) \(y^{\prime \prime}+2 y^{\prime}+y=x^{2} \cdot e^{x}+x-\cos x\)

Wir betrachten die folgende chemische Reaktion: Ein Atom vom Typ A vereinige sich mit einem Atom vom Typ \(B\) zu einem Molek?l vom Typ \(A B: A+B \rightarrow A B\). Die Anzahl der Atome vom Typ \(A\) bzw. \(B\) betrage zu Beginn der Reaktion (d.h. zur Zeit \(t=0\) ) \(a\) bzw. \(b\). Nach der Zeit \(t\) seien \(x=x(t)\) Moleküle \(A B\) entstanden. Dann läßt sich die chemische Reaktion durch die Differentialgleichung \(1 .\) Ordnung $$ \frac{d x}{d t}=k(a-x)(b-x) $$ beschreiben (k: Konstante, vom Chemiker als Geschwindigkeitskonstante bezeichnet). a) Lösen Sie diese Differentialgleichung für \(a \neq b\) und den Anfangswert \(x(0)=0\). b) Wann kommt die Reaktion zum Stillstand (Annahme: \(a>b\) )?

Lösen Sie die folgenden Anfangswertprobleme: a) \(y^{\prime \prime}+4 y^{\prime}+5 y=0, \quad y(0)=\pi, \quad y^{\prime}(0)=0\) b) \(y^{\prime \prime}+20 y^{\prime}+64 y=0, \quad y(0)=0, \quad y^{\prime}(0)=2\) c) \(4 \ddot{x}-4 \dot{x}+x=0, \quad x(0)=5, \quad \dot{x}(0)=-1\)
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