Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine Chapter 3: Problem 3
Solve the given system of differential equations by systematic elimination. $$ \begin{aligned} &\frac{d x}{d t}=-y+t \\ &\frac{d y}{d t}=x-t \end{aligned} $$
Short Answer
Expert verified
The solution to the system is \( x(t) = c_1 e^t + c_2 e^{-t} + 2 \) and \( y(t) = -c_1 e^t + c_2 e^{-t} + t \).
Step by step solution
01
Express Derivatives
From the system of differential equations, we have two equations:1. \( \frac{d x}{d t} = -y + t \)2. \( \frac{d y}{d t} = x - t \).Our goal is to express \( y \) and \( x \) in terms of their respective derivatives.
02
Substitute and Simplify
Substitute the expression for \( y \) from the first equation into the second.- From \( \frac{d x}{d t} = -y + t \), we get \( y = -\frac{d x}{d t} + t \).- Substitute in \( \frac{d y}{d t} = x - t \): \[ \frac{d y}{d t} = x - t = \frac{d^2 x}{d t^2}, \] as \( y = -\frac{d x}{d t} + t \).
03
Solve the Second Order Equation
Now, we solve the second order differential equation:\[\frac{d^2 x}{d t^2} = x - 2t.\]The complementary function is obtained from \( \frac{d^2 x}{d t^2} = x \), which yields a characteristic equation with roots \( \lambda^2 - 1 = 0 \). This gives a solution of:\[x_h(t) = c_1 e^t + c_2 e^{-t}.\]For the particular integral of \( x - 2t \), assume \( x_p = A + Bt \). Substituting gives \( A = 2 \) and \( B = 0 \). Thus,\[x_p(t) = 2.\]The general solution is:\[x(t) = c_1 e^t + c_2 e^{-t} + 2.\]
04
Find y(t) in Terms of x(t)
Using the expression for \( y \):\[y = -\frac{d x}{d t} + t,\]we substitute \( x(t) \) into this equation.The derivative \( \frac{d x}{d t} = c_1 e^t - c_2 e^{-t} \).Substitute into the equation for \( y \):\[y(t) = -c_1 e^t + c_2 e^{-t} + t.\]
05
Final Solutions
The solutions for the system of differential equations are:\( x(t) = c_1 e^t + c_2 e^{-t} + 2 \) and \( y(t) = -c_1 e^t + c_2 e^{-t} + t \).These solutions describe the behavior of \( x \) and \( y \) over time, depending on the values of the constants \( c_1 \) and \( c_2 \), determined by initial conditions.
Unlock Step-by-Step Solutions & Ace Your Exams!
-
Full Textbook Solutions
Get detailed explanations and key concepts
-
Unlimited Al creation
Al flashcards, explanations, exams and more...
-
Ads-free access
To over 500 millions flashcards
-
Money-back guarantee
We refund you if you fail your exam.
Over 30 million students worldwide already upgrade their learning with 91影视!
Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Systematic Elimination
Systematic elimination is a technique to simplify and solve systems of differential equations.
This process involves strategically substituting variables to reduce the system to a simpler form. Typically, the goal is to first express each variable in terms of derivatives, then systematically substitute one equation into another.
This process involves strategically substituting variables to reduce the system to a simpler form. Typically, the goal is to first express each variable in terms of derivatives, then systematically substitute one equation into another.
- Identify the relationships between the variables in the given equations.
- Substitute variables to eliminate one of them, often transforming the system into a single differential equation.
- Solve the resulting simplified equation.
Second Order Differential Equations
Solving a system often involves reducing it to a second order differential equation. Such equations involve second derivatives, like \(\frac{d^2 x}{d t^2}\). These can represent phenomena like acceleration in physical contexts.
When you reduce a system to the form \(\frac{d^2 x}{d t^2} = x - 2t\), a clear method emerges:
When you reduce a system to the form \(\frac{d^2 x}{d t^2} = x - 2t\), a clear method emerges:
- The homogeneous part, \(\frac{d^2 x}{d t^2} = x\), represents the system's natural behavior without external influence.
- The particular part, given non-homogeneous terms like \(-2t\), shows how the system responds to external inputs.
Characteristic Equation
The characteristic equation is essential for finding the "homogeneous" solution part of a differential equation. It comes from assuming solutions of the form \(e^{\lambda t}\), leading directly to characteristic equations like \(\lambda^2 - 1 = 0\). Here鈥檚 how it works:
- The characteristic equation results from substituting trial solutions into the differential equation.
- Solving it yields roots that determine the form of the homogeneous solution.
- For example, \(\lambda^2 - 1 = 0\) gives roots \(\lambda = 1\) and \(-1\), leading to solutions \(c_1 e^t + c_2 e^{-t}\).
General Solution
The general solution of a differential equation combines both the homogeneous solution and a particular solution. This captures the complete behavior of the system over time.
For instance, in our example,
For instance, in our example,
- The homogeneous solution \(x_h(t) = c_1 e^t + c_2 e^{-t}\) reflects natural dynamics.
- A particular solution takes into account constant external forces, like \(x_p(t) = 2\).
