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Mobile App Chapter 5: Problem 16
Lösen Sie die inhomogene lineare Differentialgleichung 1. Ordnung \(y^{\prime}-3 y=x \cdot \mathrm{e}^{x}\) a) durch Variation der Konstanten, b) durch Aufsuchen einer partikulären Lösung.
Short Answer
Expert verified
The general solution is \( y(x) = Ce^{3x} + \frac{x^2}{2} \, e^{-2x} \) .
Step by step solution
01
Solve the Homogeneous Equation
First, solve the corresponding homogeneous differential equation:\[ y^{\text{prime}} - 3y = 0 \]This is a separable differential equation. Rewrite it as:\[ \frac{dy}{dx} = 3y \]Separate variables:\[ \frac{dy}{y} = 3 \, dx \]Integrate both sides:\[ \int \frac{1}{y} \, dy = MARK: Integral_{1}^{3x} \, dx \]\[ \ln|y| = 3x + C \]Solve for y:\[ y = C \, e^{3x} \]
02
Variation of Constants
To solve the inhomogeneous equation using variation of constants, let the constant be a function of x, say: \[ y_p = u(x)e^{3x} \]Differentiate the particular solution:\[ y_p^{\text{prime}} = u'(x)e^{3x} + 3u(x)e^{3x} \]Substitute into the original differential equation:\[ u'(x)e^{3x} + 3u(x)e^{3x} - 3(u(x)e^{3x}) = x \, e^x \]Simplify:\[ u'(x) \, e^{3x} = x \, e^x \]Solve for \( u'(x) \) by dividing both sides by \( e^{3x} \):\[ u'(x) = x \, e^{-2x} \]Integrate both sides to find \( u(x) \):\[ u(x) = int_{1^{-2x}} x \, e^{-2x} \, dx \] (integrating by parts)
03
Find Partikuläre Lösung (Particular Solution)
To find a particular solution, try the method of undetermined coefficients. Assume a particular solution of the form:\[ y_p = Ax \, e^x + B \, e^x \]Take the derivative:\[ y_p^{\text{prime}} = Ae^x + Ax \, e^x + Be^x \]Substitute into the original equation:\[ (Ae^x + Ax \, e^x + Be^x) - 3(Ax \, e^x + Be^x) = x \, e^x \]Group common terms and solve for A and B:\[ Ae^x - 2Ax \, e^x -2Be^x = x \, e^x \]Compare coefficients:For \( e^x \) : \[ A = -2Ax = \]\[ -2B \]Answer: B= 0.5}
04
Sum of Solutions
The general solution to the differential equation is the sum of the homogeneous and particular solutions:\[ y(x) = y_h + y_p = Ce^{3x} + (x^2 /2)e^{-2x} \]
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Homogeneous Differential Equations
A homogeneous differential equation is one where all terms can be combined to give zero when the function and its derivatives are substituted into it. In simple terms, an equation of the form \[ y^{\text{prime}} + p(x) y = 0 \] is homogeneous.Understanding how to solve these equations is useful because often, solving their inhomogeneous counterparts requires us to first solve the homogeneous version.To solve a homogeneous differential equation:
- Rewrite the equation so that it's easy to separate variables.
- Separate the variables and integrate both sides.
- Solve for the function, typically resulting in an exponential function.
Variation of Constants
Variation of constants (or parameters) is a method used to solve inhomogeneous linear differential equations. Instead of having a constant coefficient like in homogeneous solutions, we allow it to vary.For the given problem, we determined from the homogeneous solution that \[ y = Ce^{3x}.\] Instead of a constant C, we let it become a function: \[ y_p = u(x)e^{3x}.\] By differentiating this new form and substituting into the original inhomogeneous equation, we simplify and solve for the function u(x). For this problem, the differentiation leads to an expression. After substituting and simplifying, we find \[ u^{\text{prime}}(x) = x \times e^{-2x},\] which we then integrate to determine u(x). This method is particularly helpful because it leverages the structure of the homogeneous solution while accommodating the inhomogeneity.
Particular Solution
Finding a particular solution to an inhomogeneous differential equation involves guessing a function form that satisfies the non-homogeneous term. This is essential because the solution to an inhomogeneous differential equation is the sum of the homogeneous solution and a particular solution. For the given equation, \[y^{\text{prime}} - 3y = x \times e^x,\] we propose a particular solution of the form \[y_p = A x e^x + B e^x.\] By differentiating this form and substituting it back into the original differential equation, we identify the coefficients A and B that satisfy the equation. This approach greatly simplifies finding the particular solution and does not require knowing the general form from the start.
Undetermined Coefficients
The method of undetermined coefficients is a strategy for finding the particular solution of a linear inhomogeneous differential equation when the non-homogeneous term is of a certain type. These types typically include polynomials, exponentials, sines, and cosines. For our problem \[y^{\text{prime}} - 3y = x \times e^x,\] we guess a particular solution of the form \[y_p = A x e^x + B e^x.\]By differentiating and substituting it back into the original equation, we solve for the constants A and B. Here, undetermined coefficients work well because our guessed solution form matches the form of the non-homogeneous term.The steps involve
- Guessing the form of the solution,
- Taking derivatives as needed,
- Substituting back into the differential equation,
- Matching coefficients to solve for any unknowns.
