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Chapter 5: Problem 26

Zeigen Sie, daB sich die nicht-lineare Differentialgleichung 1. Ordnung $$ 4 y y^{\prime}-y^{2}=-\left(1+x^{2}\right) $$ mit Hilfe der Substitution \(u=y^{2}\) in eine lineare Differentialgleichung 1 . Ordnung überführen läBt und bestimmen Sie die allgemeine Lösung dieser Differentialgleichung.

Short Answer

Expert verified
Substitute \( y^2 = u \), solve the linear DE \( 2 \frac{du}{dx} + u = 1 + x^2 \). Find \( y = \sqrt{u} \).

Step by step solution

01

Substitute variable

Let the substitution be given as \( u = y^2 \). This implies \( y = \sqrt{u} \).
02

Compute the derivative

Given \( u = y^2 \), differentiate both sides with respect to \( x \): \( \frac{du}{dx} = \frac{d}{dx}(y^2) \). By the chain rule, \( \frac{du}{dx} = 2yy' \). Thus, \( yy' = \frac{1}{2} \frac{du}{dx} \).
03

Substitution into original equation

Substitute for \( y \) and \( yy' \) in the original equation: \( 4yy' - y^2 = -(1 + x^2) \) becomes \( 4 \frac{1}{2} \frac{du}{dx} - u = -(1 + x^2) \).
04

Simplify the equation

Simplify the substituted equation: \( 2 \frac{du}{dx} - u = -(1 + x^2) \) Add \( u \) to both sides: \( 2 \frac{du}{dx} = u - (1 + x^2) \).
05

Rearrange into standard linear form

The differential equation can be written in standard linear form: \( 2 \frac{du}{dx} + u = 1 + x^2 \).
06

Solve the linear differential equation

The general solution can be obtained using the integrating factor method. The linear equation is of the form \( \frac{du}{dx} + P(x)u = Q(x) \), where \( P(x) = \frac{1}{2} \) and \( Q(x) = \frac{1 + x^2}{2} \). The integrating factor is \( e^{\frac{x}{2}} \). Multiply through by the integrating factor and solve.
07

Integrate and find the solution

After multiplying by \( e^{\frac{x}{2}} \), we integrate both sides to find: \( u = e^{-\frac{x}{2}} \int (1 + x^2)e^{\frac{x}{2}} dx + C e^{-\frac{x}{2}} \) where \( C \) is an integration constant. Compute the integral and simplify.
08

Back-substitute to find \( y \)

Replace \( u \) with \( y^2 \): \( y^2 = e^{-\frac{x}{2}} \int (1 + x^2)e^{\frac{x}{2}} dx + C e^{-\frac{x}{2}} \). \( y \) is then the square root of this expression.

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Key Concepts

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

First-Order Differential Equations
First-order differential equations are equations that involve the derivative of an unknown function and the function itself. These equations are commonly expressed in the form \( \frac{dy}{dx} = f(x, y) \). The key to solving them often lies in identifying the right method, such as separation of variables, integrating factors, or substitution. These equations are fundamental as they describe various real-world phenomena, from population growth to electrical circuits. Understanding first-order differential equations provides a solid foundation for tackling more complex differential equations.
Substitution Method
The substitution method is a powerful tool for simplifying differential equations. By introducing a new variable, we can transform a difficult equation into a more straightforward form. For the given problem, we used the substitution \( u = y^2 \). This leads to \( yy' = \frac{1}{2} \frac{du}{dx} \). By replacing these expressions in the original differential equation, we turned it into a linear form: \( 2 \frac{du}{dx} + u = 1 + x^2 \). This new equation is simpler to solve using standard methods. The essence of the substitution method is to reduce complexity and make the equation manageable. It's widely used across different areas of science and engineering.
Linearization
Linearization involves transforming a non-linear equation into a linear one. Linear equations are easier to solve due to their simpler structure. In the given exercise, after substitution, we rearranged the terms to obtain a linear differential equation: \( 2 \frac{du}{dx} + u = 1 + x^2 \). This step is crucial as it allows us to apply well-known techniques for linear equations, such as the integrating factor method. Linearization is an essential method in mathematical modeling, helping to approximate and solve complex non-linear systems by treating them as linear within a certain range.

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Most popular questions from this chapter

Das Anfangswertproblem $$ \ddot{x}+4 \dot{x}+29 x=0, \quad x(0)=1, \quad \dot{x}(0)=v(0)=-2 $$ beschreibt eine ged?mpfte (mechanische) Schwingung \((x:\) Auslenkung; \(v=\dot{x}:\) Geschwindigkeit). Wie groB sind Auslenkung und Geschwindigkeit zur Zeit \(t=0,1\) a) bei exakter Lösung, b) bei näherungsweiser Lösung der Differentialgleichung nach dem Runge- KuttaVerfahren 4. Ordnung (Schrittweite: \(h=\Delta t=0,05\) ).

Zeigen Sie: Die Funktionen $$ y_{1}(x)=\mathrm{e}^{2 x} \quad \text { und } \quad y_{2}(x)=x \cdot \mathrm{e}^{2 x} $$ bilden eine Fundamentalbasis der homogenen Differentialgleichung 2 . Ordnung \(y^{\prime \prime}-4 y^{\prime}+4 y=0\)

Lösen Sie die folgenden Anfangswertprobleme: a) \(x^{2} y^{\prime}=y^{2}+x y, \quad y(1)=-1\) b) \(y y^{\prime}=2 \cdot \mathrm{e}^{2 x}, \quad y(0)=2\)

Bestimmen Sie die allgemeine Lösung der folgenden in der Matrizenform dargestellten Systeme homogener linearer Differentialgleichungen 1. Ordnung: a) \(\left(\begin{array}{l}y_{1}^{\prime} \\\ y_{2}^{\prime}\end{array}\right)=\left(\begin{array}{rr}4 & -4 \\ 1 & 8\end{array}\right)\left(\begin{array}{l}y_{1} \\ y_{2}\end{array}\right)\) b) \(\left(\begin{array}{l}\dot{x}_{1} \\\ \dot{x}_{2}\end{array}\right)=\left(\begin{array}{rr}2 & -1 \\ -4 & 2\end{array}\right)\left(\begin{array}{l}x_{1} \\ x_{2}\end{array}\right)\) c) \(\left(\begin{array}{l}y_{1}^{\prime} \\\ y_{2}^{\prime}\end{array}\right)=\left(\begin{array}{rr}-1 & -1 \\ 2 & -3\end{array}\right)\left(\begin{array}{l}y_{1} \\ y_{2}\end{array}\right)\)

Ein schwingungsfahiges mechanisches System bestehe aus einer Masse \(m=0,5 \mathrm{~kg}\) und einer Feder mit der Federkonstanten \(c=128 \mathrm{~N} / \mathrm{m}\). a) Wie groB \(\mathrm{muB}\) die Dämpferkonstante \(b\) sein, damit gerade der aperiodische Grenzfall eintritt? Für welche Werte von \(b\) schwingt das System aperiodisch? b) Lösen Sie die Schwingungsgleichung für dea unter (a) behandelten aperiodischen Grenzfall, wenn zu Beginn der Bewegung gilt: \(x(0)=0,2 \mathrm{~m}\), \(v(0)=0 .\) Skizzieren Sie den „Schwingungsverlauf".
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