What will be the required solution of d^2y/dx^2 – 3dy/dx + 4y = 0?

Category: QuestionsWhat will be the required solution of d^2y/dx^2 – 3dy/dx + 4y = 0?
Editor">Editor Staff asked 11 months ago

What will be the required solution of d^2y/dx^2 – 3dy/dx + 4y = 0?
 
(a) Ae^-4x + Be^-x
 
(b) Ae^4x – Be^-x
 
(c) Ae^4x + Be^-x
 
(d) Ae^4x + Be^x
 
I had been asked this question by my school principal while I was bunking the class.
 
This key question is from Linear Second Order Differential Equations topic in division Differential Equations of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
Interview Questions and Answers, Database Interview Questions and Answers for Freshers and Experience

1 Answers
Editor">Editor Staff answered 11 months ago

The correct choice is (c) Ae^4x + Be^-x
 
To elaborate: d^2y/dx^2 – 3dy/dx + 4y = 0  …..(1)
 
Let, y = e^mx be a trial solution of (1), then,
 
=> dy/dx = me^mx and d^2y/dx^2 = m^2e^mx
 
Clearly, y = e^mx will satisfy equation (1). Hence, we have,
 
m^2e^mx – 3m * e^mx – 4e^mx = 0
 
=>m^2 – 3m – 4 = 0 (as e^mx ≠ 0)   …….(2)
 
=> m^2 – 4m + m – 4 = 0
 
=> m(m – 4) + 1(m – 4) = 0
 
Or, (m – 4)(m + 1) = 0
 
Thus, m = 4 or m = -1
 
Clearly, the roots of the auxiliary equation (2) are real and unequal.
 
Therefore, the required general solution of (1) is
 
y = Ae^4x + Be^-x where A and B are constants.