How do I find the interval for solving this differential equation?

  • Context: MHB 
  • Thread starter Thread starter karush
  • Start date Start date
  • Tags Tags
    Interval
Click For Summary
SUMMARY

The discussion focuses on finding the interval for solving the differential equation \(y' + \frac{\ln{t}}{t-3}y=\frac{2t}{t-3}\). Participants identify that the denominator \(t-3\) indicates that \(t\) cannot equal 3, while the term \(\ln(t)\) necessitates that \(t > 0\). The conclusion is that the valid interval for \(t\) is \( (0, 3) \) due to these restrictions.

PREREQUISITES
  • Understanding of first-order linear differential equations
  • Knowledge of logarithmic functions, specifically \(\ln(t)\)
  • Familiarity with the method of integrating factors
  • Basic calculus concepts, including integration and limits
NEXT STEPS
  • Study the method of integrating factors in differential equations
  • Learn about the properties and applications of logarithmic functions
  • Research the implications of singular points in differential equations
  • Explore the concept of intervals of existence for solutions to differential equations
USEFUL FOR

Mathematics students, educators, and anyone studying differential equations who seeks to understand the conditions for solution intervals and the implications of logarithmic terms in equations.

karush
Gold Member
MHB
Messages
3,240
Reaction score
5
View attachment 8690
doing #1

ok first I divided thru

$$y' + \frac{\ln{t}}{t-3}y=\frac{2t}{t-3}$$

but the $$\exp\int p(t) \, dt$$ step kinda baloated?

ok I see the denominator has $t-3$ so presume 3 is one of the interval ends
but why 0. ?

book answwer is

View attachment 8691
 

Attachments

  • 2.4.1.PNG
    2.4.1.PNG
    7.7 KB · Views: 149
  • 2.4.1-6a.PNG
    2.4.1-6a.PNG
    2.5 KB · Views: 134
Physics news on Phys.org
"Baloated"?

The reason for the restriction x> 0 is the "ln(x)" term.
 
Country Boy said:
"Baloated"?

The reason for the restriction x> 0 is the "ln(x)" term.

you probably mean $ln{t}$

yeah how would this really work $e^{p}$
View attachment 8693
 
I am not sure what you are asking. Do you know what "$Li_2(x)$" is?
 
karush said:
yeah how would this really work $e^{p}$
But your problem statement said not to solve the equation. So why are you introducing the [math]e^p[/math] into your solution?

-Dan
 

Similar threads

MHB 2.1.2 Find the general solution of the given differential equation
  • · Replies 2 ·
Replies
2
Views
1K
MHB -b.1.1.2 behavior of y'-2y=-3 as t goes to infinity
  • · Replies 3 ·
Replies
3
Views
1K
MHB 2.3.17 Find the average acceleration for this interval.
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Determining equilibrium solutions to differential equations
  • · Replies 16 ·
Replies
16
Views
3K
MHB 2.1.2 Find the general solution of y'−2y=t^{2}e^{2t}
  • · Replies 4 ·
Replies
4
Views
4K
MHB -b.1.1.4 directional field as t \to \infny
  • · Replies 2 ·
Replies
2
Views
2K
MHB -2.1.9 Find general solution of 2y'+y=3t
  • · Replies 7 ·
Replies
7
Views
3K
MHB -b.2.1.5 Find the general solution of y' - 2y =3te^t,
  • · Replies 2 ·
Replies
2
Views
2K
MHB -2.1.6 ty'- 2y =\sin{t}, t>0
  • · Replies 7 ·
Replies
7
Views
3K
MHB 2.1.7 DE y'e -2ty =2te^(-t^2)
  • · Replies 3 ·
Replies
3
Views
2K