What is the Behavior of Solutions of a DE with Limits as t Approaches Infinity?

  • Thread starter Thread starter ehabmozart
  • Start date Start date
  • Tags Tags
    Limits
Click For Summary
SUMMARY

The general solution to the differential equation y' - 2y = 3e^t is y = -3e^t + ce^(2t). As t approaches infinity, the term ce^(2t) dominates due to its exponential growth compared to -3e^t. Therefore, if c is positive, the solution approaches positive infinity, while if c is negative, the solution approaches negative infinity. This confirms that all solutions will increase exponentially as t approaches infinity.

PREREQUISITES
  • Understanding of differential equations, specifically first-order linear DEs.
  • Knowledge of exponential functions and their limits.
  • Familiarity with the concept of general solutions in differential equations.
  • Basic calculus, particularly limits and behavior of functions as they approach infinity.
NEXT STEPS
  • Study the behavior of solutions to linear differential equations with constant coefficients.
  • Learn about the method of integrating factors for solving first-order DEs.
  • Explore the implications of initial conditions on the general solution of differential equations.
  • Investigate the stability of solutions in the context of differential equations.
USEFUL FOR

Students studying differential equations, mathematicians interested in the behavior of solutions, and educators teaching calculus and differential equations.

ehabmozart
Messages
212
Reaction score
0

Homework Statement



Find the general solution of the given differential equation, and use it to determine how
solutions behave as t→∞.

y' − 2y = 3et



Homework Equations



DE

The Attempt at a Solution



After some work, I got y=-3et+ce2t . Now I have problems in getting the limit as t goes to infinity. C can possibly be a positive or negative value. In case it is -ve, the answer goes to negative infinity. If it is positive, I can't really figure out what would the limit be. In the book however, it is written that 'It follows
that all solutions will increase exponentially'. HOW?
 
Physics news on Phys.org
e2t will beat et, so it will go to positive or negative infinity according to the sign of c.
 

Similar threads

Solve DE Involving Limits: y' - 2y = t^2e^2t
  • · Replies 8 ·
Replies
8
Views
2K
Help Taking the Limit as K goes to infinity
  • · Replies 14 ·
Replies
14
Views
3K
Prove that solutions to autonomous first order DE can't have local maxima
  • · Replies 2 ·
Replies
2
Views
1K
Behavior of DE as t approaches infinity
  • · Replies 5 ·
Replies
5
Views
5K
Obtain a nonzero solution of ##y''-4y'+x^2(y'-4y)=0## by inspection
  • · Replies 7 ·
Replies
7
Views
2K
Limit as x approaches infinity.
  • · Replies 2 ·
Replies
2
Views
2K
Determining the limit for function of x and y
  • · Replies 1 ·
Replies
1
Views
1K
Evaluating Limits: x Approach 0 & Beyond
  • · Replies 3 ·
Replies
3
Views
1K
Slope Fields and Determing Behavior of Any Solution
  • · Replies 2 ·
Replies
2
Views
2K
Finding k in a probability density function
  • · Replies 4 ·
Replies
4
Views
4K