JEE 2023 — Mathematics PYQ
JEE | Mathematics | 2023Let y=y(t) be a solution of the differential equation dtdy+αy=γe−βt where, \alpha > 0, \beta > 0 and \gamma > 0. Then limt→∞y(t)
Choose the correct answer:
- A.
is −1
- B.
is 1
- C.
does not exist
- D.
is 0
(Correct Answer)
is 0
Explanation
Solving
1. Identify the Equation Type
This is a Linear Differential Equation of the form dtdy+Py=Q.
-
P=α
-
Q=γe−βt
2. Integrating Factor (IF)
3. General Solution
4. Evaluate for α=β
Divide by eαt:
5. Apply Limit (t→∞)
Given \alpha > 0 and \beta > 0:
6. Evaluate for α=β
Using L'Hôpital's Rule for the first term:
Thus, limt→∞y(t)=0.
Correct Option: (4)
Explanation
Solving
1. Identify the Equation Type
This is a Linear Differential Equation of the form dtdy+Py=Q.
-
P=α
-
Q=γe−βt
2. Integrating Factor (IF)
3. General Solution
4. Evaluate for α=β
Divide by eαt:
5. Apply Limit (t→∞)
Given \alpha > 0 and \beta > 0:
6. Evaluate for α=β
Using L'Hôpital's Rule for the first term:
Thus, limt→∞y(t)=0.
Correct Option: (4)

