ZaidAlyafey said:What is \(\displaystyle w\) ? ,is it a constant or a function of $t$ ?
krish said:I believe w is a constant.
ZaidAlyafey said:Aha , try differentiating one of the equations and tell me if you got any ideas .
krish said:So I differentiated the second equation with respect to t:
v'' = -du/dt
Then I substitute first equation for du/dt:
v'' = -(v - w(t-5)) = -v + w(t-5)
v'' + v - w(t-5) = 0
Does it become: v'' + v = wt - 5w ? How does keeping the w matter?
And then I solve the IVP. Is this correct? But what if w is a function of t? Then I am confused, is that possible? Thank you for answering.
ZaidAlyafey said:That seems a not-easy problem to deal with if we try the other differentiation we get
\(\displaystyle u''+ut = 2- w \)
The problem will get more complicated if assumed that $w$ a function because we have a three functions and two equations !
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It might be solvable by Laplace but I don't know whether it is an acceptable solution ?
krish said:Can w(t-5) be the Heavyside unit function?
ZaidAlyafey said:I don't know there is no indication , that depends on the source.
From Where did you get that problem ?
krish said:It's in the review questions in my Differential Equations textbook.
ZaidAlyafey said:Ok , tell me the name of the textbook and the page number .
An initial value problem (IVP) for a system of ordinary differential equations (ODEs) involves finding a solution that satisfies a set of differential equations at a given initial condition. This means that the solution must satisfy the equations at a specific starting point, typically denoted as t = 0.
The most common method for solving an IVP for a system of ODEs is using numerical methods. This involves approximating the solution at discrete points in time and using algorithms such as Euler's method or the Runge-Kutta method to find the solution. Other methods such as Laplace transforms or separation of variables may also be used for specific types of ODEs.
One of the main challenges in solving an IVP for a system of ODEs is the complexity of the equations. Depending on the system, it may be difficult to find an analytical solution and numerical methods may be necessary. Another challenge is ensuring the numerical solution is accurate and does not have significant errors due to rounding or other factors.
Yes, an IVP for a system of ODEs can have multiple solutions. This can occur when the equations are not well-defined or if there are multiple sets of initial conditions that satisfy the equations. In some cases, a system of ODEs may have an infinite number of solutions.
Yes, solving an IVP for a system of ODEs has many real-world applications. For example, it is commonly used in physics to model the behavior of physical systems such as pendulums or electric circuits. It is also used in engineering for designing and analyzing systems such as control systems or chemical reactions. Additionally, it is used in economics for modeling population growth or stock prices.