- #1
First, what is your c1 in terms of \(\displaystyle y_0\)? You never finished that part.karush said:$\quad\displaystyle \frac{dy}{dt}=2y-5, \quad y(0)=y_0$
rewrite
$\quad y'-2y=-5$
obtain u(x)
$\quad u(x)=\exp\int-2\, dx = e^{-2t}$
then
$\quad (e^{-2t}y)'=-5e^{-2t}$
integrate both sides
$\quad\displaystyle e^{-2t}y=-5\int e^{-2t} dt=-5e^{-2t}+c_1$
finally
$\quad\displaystyle y
=-5\frac{e^{-2t}}{e^{-2t}}+\frac{c_1}{e^{-2t}}=c_1e^{2t}-5$
so at $\quad y(0)=y_0$
ok I don't see how they got this answer
well that is pretty cool ... better than the book processMarkFL said:We have:
\(\displaystyle \frac{d}{dt}\left(e^{-2t}y\right)=-5e^{-2t}\)
Integrate using the boundaries:
\(\displaystyle \int_{y_0}^{e^{-2t}y}\,du=-5\int_{0}^{t}e^{-2v}\,dv\)
\(\displaystyle e^{-2t}y-y_0=\frac{5}{2}\left(e^{-2t}-1\right)\)
\(\displaystyle y-y_0e^{2t}=\frac{5}{2}\left(1-e^{2t}\right)\)
\(\displaystyle y=\left(y_0-\frac{5}{2}\right)e^{2t}+\frac{5}{2}\)
When solving differential equations you see a lot of arbitrary constants. (ie. there are many solutions to the same differential equation.) The boundary conditions simply give you information about those constants. In this case you had c1 as a constant after solving the differential equation. The boundary condition \(\displaystyle y(0) = y_0\) tells you what c1 is in terms of \(\displaystyle y_0\). Perhaps it would be more meaningful to you if we specified y(0) = 3 or something like that.karush said:well that is pretty cool ... better than the book process
but I still don't think the boundary thing has registered with me
An initial value problem is a type of mathematical problem that involves finding a function or set of functions that satisfy a given set of conditions, typically at a specific point or set of points. These conditions typically include an initial value, such as a starting point or boundary condition, and a set of differential equations or other equations that describe the behavior of the function.
The "-b.1.2.2c" notation refers to a specific type of initial value problem that is used in mathematics and physics. This notation typically indicates that the problem involves a second-order ordinary differential equation with constant coefficients.
The purpose of solving an initial value problem is to determine the behavior of a function or set of functions at specific points or over a specific range of values. This can be useful in modeling physical systems, predicting future behavior, and understanding the underlying principles and relationships that govern a system.
Some common methods for solving "-b.1.2.2c initial value problems" include the Laplace transform method, the method of undetermined coefficients, and the variation of parameters method. These methods involve different techniques for manipulating and solving differential equations to find the desired functions.
"-b.1.2.2c initial value problems" have many applications in the real world, including in physics, engineering, and economics. They can be used to model physical systems, such as the motion of objects or the flow of fluids, and to predict behavior in these systems. They are also commonly used in economic models to predict future trends and make informed decisions.