-2.2.27 - Analysis of first order IVP

In summary, you need to find the equation of the tangent line to the graph at $y=4$ and use C to find the slope.
  • #1
karush
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well each one is a little different so,,,
$$\dfrac{dy}{dt}=\dfrac{ty(4-y)}{3},\qquad y(0) =y_0$$
not sure if this is what they meant on the given expression
 
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  • #2
Could you be clear on what you're struggling with? For instance a) requires first the solution of the DE. Can you identify the kind of DE that it is and suggest a solution approach? (e.g., separable)
 
  • #3
You just need to integrate $$dy/(y(4-y))=tdt/3$$ by using the partial fractions $1/(y(4-y))=A/y+B/(4-y)$ where you find A and B by multiplying first by $y$ and then plugging y=0 to find A, as for B just try to multiply by 4-y and again plug $y=4$ to find the coefficients A and B. Thus you immediate integrals to solve.
 
  • #4
Joppy said:
Could you be clear on what you're struggling with? For instance a) requires first the solution of the DE. Can you identify the kind of DE that it is and suggest a solution approach? (e.g., separable)
ok yes the 2,2 section exercises are all on separable equations
I have 10 more problems I want to post here
I'm retired and by myself on this. there is no real need for me me even for me to do this but it certainly has given me a sense of accomplishment
MHB has been a great place for me to spend all the free time I have
 
  • #5
Yes, I see what looks like a separable first order ODE for which partial fractions will prove to be useful. :)
 
  • #6
Alan said:
You just need to integrate $$dy/(y(4-y))=tdt/3$$ by using the partial fractions $1/(y(4-y))=A/y+B/(4-y)$ where you find A and B by multiplying first by $y$ and then plugging y=0 to find A, as for B just try to multiply by 4-y and again plug $y=4$ to find the coefficients A and B. Thus you immediate integrals to solve.
$\dfrac{1}{(4 - y) y} = \dfrac{ 1} {4 y} - \dfrac{1}{4 (y - 4)}$
like this?
do we really need this ugly brown box at the bottom!
 
  • #7
karush said:
$\dfrac{1}{(4 - y) y} = \dfrac{ 1} {4 y} - \dfrac{1}{4 (y - 4)}$
like this?
do we really need this ugly brown box at the bottom!
You are off by a minus sign.

\(\displaystyle \dfrac{1}{y(y - 4)} = \dfrac{1}{4y} - \dfrac{1}{4(y - 4)}\)

Watch the negative sign on the 4 - y.

-Dan
 
  • #8
$\displaystyle \dfrac{1}{y(4¬y)} = \dfrac{1}{4y} - \dfrac{1}{4(y - 4)}$
thus
if so then
$$\dfrac{1}{4} \displaystyle\int \dfrac{1}{y} \, dy- \dfrac{1}{4} \int\dfrac{1}{y-4}\, dy
=\dfrac{1}{3} \int t\, dt $$
 
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  • #9
karush said:
$\displaystyle \dfrac{1}{y(4¬y)} = \dfrac{1}{4y} - \dfrac{1}{4(y - 4)}$
thus
if so then
$$\dfrac{1}{4} \displaystyle\int \dfrac{1}{y} \, dy- \dfrac{1}{4} \int\dfrac{1}{y-4}\, dy
=\dfrac{1}{3} \int t\, dt $$

how about ...

$$\displaystyle\int \dfrac{1}{y} \, dy- \int\dfrac{1}{y-4}\, dy
=\dfrac{4}{3} \int t\, dt $$
 
  • #10
This is also a Bernoulli equation, if you wanted to take a different route :). But I'm not sure if your book covered linear ODE's before separable.
 
  • #11
karush said:
$\dfrac{1}{(4 - y) y} = \dfrac{ 1} {4 y} - \dfrac{1}{4 (y - 4)}$
like this?
do we really need this ugly brown box at the bottom!
Which ugly box are you talking about?
 
  • #12
Alan said:
Which ugly box are you talking about?
on a tablet the brown footer is a huge retangle with ears on the sides
terrifying to stare at
 
  • #13
$$\displaystyle\int \dfrac{1}{y} \, dy- \int\dfrac{1}{y-4}\, dy
=\dfrac{4}{3} \int t\, dt$$

$$\ln |y|-\ln |y-4|=\frac{2t^2}{3}+C$$

ok assume find C from RHS and $y(0) =y_0$

does $y_0=t$??
 
Last edited:
  • #14
karush said:
on a tablet the brown footer is a huge retangle with ears on the sides
terrifying to stare at
OK, I am using a Desktop and this ugly box isn't shown there.
 
  • #15
karush said:
$$\displaystyle\int \dfrac{1}{y} \, dy- \int\dfrac{1}{y-4}\, dy
=\dfrac{4}{3} \int t\, dt$$

$$\ln |y|-\ln |y-4|=\frac{2t^2}{3}+C$$

ok assume find C from RHS and $y(0) =y_0$

does $y_0=t$??
y_0 is a constant and t is a variable, so this doesn't make so much sense.
 
  • #16
2020_05_06_10.28.47~2.jpg

OK here is the book answer. #27 just not sure how it was derived!
 

FAQ: -2.2.27 - Analysis of first order IVP

1. What is a first order initial value problem (IVP)?

A first order initial value problem is a type of differential equation that involves finding a function that satisfies both a given differential equation and a specified initial condition. The initial condition typically involves a value for the function at a specific point.

2. How is a first order IVP solved?

A first order IVP is typically solved using analytical or numerical methods. Analytical methods involve finding an exact solution to the differential equation, while numerical methods involve approximating the solution using a computer algorithm.

3. What is the difference between an explicit and implicit solution to a first order IVP?

An explicit solution to a first order IVP is one where the dependent variable (usually denoted by y) is isolated on one side of the equation. An implicit solution, on the other hand, is one where the dependent variable appears on both sides of the equation.

4. What is a particular solution to a first order IVP?

A particular solution to a first order IVP is a specific solution that satisfies both the differential equation and the initial condition. This solution is unique for a given initial condition, while the general solution may have multiple solutions.

5. How are first order IVPs used in real-world applications?

First order IVPs are used to model a wide range of real-world phenomena, including population growth, chemical reactions, and electrical circuits. They are also used in physics and engineering to describe the behavior of systems over time.

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