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Chapter 5: Problem 9

Welche allgemeinen L?sungen besitzen die folgenden inhomogenen linearen Differentialgleichungen 4 . und 5 . Ordnung: a) \(y^{(4)}+2 y^{\prime \prime}+y=8 \cdot \sin x+x^{2}+4\) b) \(y^{(5)}+3 y^{(4)}+3 y^{\prime \prime \prime}+y^{\prime \prime}=2(\sin x+\cos x+1)\)

Short Answer

Expert verified
The general solutions are:(a) \( y = C_1 \text{cos}(x) + C_2 \text{sin}(x) + C_3 x \text{cos}(x) + C_4 x \text{sin}(x) + 8 \text{sin}(x) + x^2 + 4 \).(b) \( y = C_1 + C_2 e^{-x} + C_3 e^{ix} + C_4 e^{-ix} + C_5 x + A \text{sin}(x) + B \text{cos}(x) + C \).

Step by step solution

01

Identify the Homogeneous Part of the Differential Equation

Separate the homogeneous part of each differential equation from the inhomogeneous part. For equation (a): \( y^{(4)}+2 y^{\text{''}}+y=0 \). For equation (b): \( y^{(5)}+3 y^{(4)}+3 y^{\text{''}'}+y^{\text{''}}=0 \).
02

Solve the Characteristic Equation for the Homogeneous Part

Determine the characteristic equation for each homogeneous differential equation. For equation (a): \( r^4 + 2r^2 + 1 = 0 \), which simplifies to \( (r^2 + 1)^2 = 0 \) giving roots \( r = i, -i \). For equation (b): \( r^5 + 3r^4 + 3r^3 + r^2 = 0 \), which simplifies to \( r^2(r^3 + 3r^2 + 3r + 1) = 0 \) giving roots \( r = 0 \) and solving the cubic factor yields complex roots.
03

Form the General Solution for the Homogeneous Part

Construct the general solution from the roots of the characteristic equation. For equation (a): \( y_h = C_1 \text{cos}(x) + C_2 \text{sin}(x) + C_3 x \text{cos}(x) + C_4 x \text{sin}(x) \). For equation (b): \( y_h = C_1 + C_2 e^{-x} + C_3 e^{ix} + C_4 e^{-ix} + C_5 x \).
04

Solve the Inhomogeneous Part Using the Method of Undetermined Coefficients

Propose a particular solution for the inhomogeneous part. For equation (a) propose: \( y_p = A \text{sin}(x) + B x^2 + C x + D \). For equation (b) propose: \( y_p = A \text{sin}(x) + B \text{cos}(x) + C \).
05

Determine the Coefficients for the Particular Solution

Substitute the proposed particular solution into the original differential equation and solve for the coefficients. For equation (a), \( A = 8 \) for the \( \text{sin}(x) \) term, and \( B, C, D \) for the polynomial \( x^2 + 4 \). For equation (b), solve for coefficients \( A, B, C \) to satisfy \( 2(\text{sin}(x) + \text{cos}(x) + 1) \).
06

Combine the Homogeneous and Particular Solutions

Add the homogeneous and particular solutions together to obtain the general solution. For equation (a): \( y(x) = C_1 \text{cos}(x) + C_2 \text{sin}(x) + C_3 x \text{cos}(x) + C_4 x \text{sin}(x) + 8 \text{sin}(x) + x^2 + 4 \). For equation (b): \( y(x) = C_1 + C_2 e^{-x} + C_3 e^{ix} + C_4 e^{-ix} + C_5 x + A \text{sin}(x) + B \text{cos}(x) + C \).

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Key Concepts

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

Characteristic Equation
When you encounter a differential equation, the first step is to deal with the homogeneous part. This part doesn't include the constants or terms with only functions.
For example, consider the equation: \( y^{(4)} + 2 y'' + y = 8 \sin x + x^2 + 4 \).
Method of Undetermined Coefficients
Once the homogeneous equation is solved, we need to find a particular solution to the non-homogeneous equation.
The idea of the method of undetermined coefficients is to suggest a form for the solution based on the non-homogeneous term(s).
For example, if we have \( y^{(4)} + 2 y^{\text{''}} + y = 8 \sin x + x^2 + 4 \), we suggest \( y_p = A \sin x + B x^2 + C x + D \).
Here, \(y_p\) is the particular solution.
General Solution
The last part is combining the solutions.
We integrate the particular solution and the general solution of the homogeneous equation to get the full solution.
For example, the general solution to \( y^{(4)} + 2 y'' + y = 8 \sin x + x^2 + 4 \) is \( y(x) = C_1 \cos(x) + C_2 \sin(x) + C_3 x \cos(x) + C_4 x \sin(x) + 8 \sin(x) + x^2 + 4 \).
This combination includes all solutions, covering both homogeneous and non-homogeneous parts.

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Most popular questions from this chapter

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.

Bestimmen Sie die allgemeinen L?sungen der folgenden homogenen linearen Differentialgleichungen 5 . Ordnung: a) \(x^{(5)}+2 \dddot x+\dot{x}=0\) b) \(y^{(5)}+2 y^{(4)}+y^{\prime \prime \prime}=0\) c) \(y^{(5)}+3 y^{(4)}+10 y^{\prime \prime \prime}+6 y^{\prime \prime}+5 y^{\prime}-25 y=0\) d) \(y^{(5)}+22 y^{\prime \prime \prime}+2 y^{\prime \prime}-75 y^{\prime}+50 y=0\)

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 allgemeine Lösung der folgenden Systeme inhomogener linearer Differentialgleichungen 1. Ordnung nach dem Einsetzungs- oder Eliminationsverfahren: a) $$ \begin{aligned} &y_{1}^{\prime}=-2 y_{1}-2 y_{2}+\mathrm{e}^{x} \\ &y_{2}^{\prime}=5 y_{1}+4 y_{2} \end{aligned} $$ b) \(\left(\begin{array}{l}y_{1}^{\prime} \\\ y_{2}^{\prime}\end{array}\right)=\left(\begin{array}{rr}-3 & -2 \\ 2 & 1\end{array}\right)\left(\begin{array}{l}y_{1} \\\ y_{2}\end{array}\right)+\left(\begin{array}{l}3 x \\ 2 x\end{array}\right)\) c) \(\left(\begin{array}{l}\dot{x}_{1} \\\ \dot{x}_{2}\end{array}\right)=\left(\begin{array}{rr}3 & -1 \\ -1 & 3\end{array}\right)\left(\begin{array}{l}x_{1} \\\ x_{2}\end{array}\right)+8\left(\begin{array}{c}\mathrm{e}^{3 i} \\\ 1\end{array}\right)\)

Welche der folgenden linearen Differentialgleichungen \(2 .\) Ordnung besitzen konstante Koeffizienten? Klassifizieren Sie diese Differentialgleichungen weiter nach homogenen und inhomogenen Gleichungen. a) \(y^{\prime \prime}+2 y^{\prime}+y=\cos x\) b) \(\quad x y^{\prime \prime}-2 y^{\prime}=0\) c) \(y^{\prime \prime}+6 y^{\prime}+9 y=0\) d) \(2 \vec{x}+x=e^{-2 t}\) e) \(y^{\prime \prime}+y^{\prime}+x^{2} y=\mathrm{e}^{x}\) f) \(y^{\prime \prime}-4 y^{\prime}+13 y=0\)
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