The solution of differential equation will be ________, where c₁ and c₂ are constants.

Question

Question

This question was previously asked in

AAI ATC Junior Executive 2018 Official Paper (Shift 3)

Answer 1

Concept:

For different roots of the auxiliary equation, the solution (complementary function) of the differential equation is as shown below.

Roots of Auxiliary Equation	Complementary Function
m1, m2, m3, … (real and different roots)
m1, m1, m3, … (two real and equal roots)
m1, m1, m1, m4… (three real and equal roots)
α + i β, α – i β, m3, … (a pair of imaginary roots)
α ± i β, α ± i β, m5, … (two pairs of equal imaginary roots)

Calculation:

Given:

Put x = et

⇒ t = ln x

Now, the above differential equation becomes

D(D - 1)y + 4 Dy + 2y = 0 ⇒ (D² + 3D + 2)y = 0

⇒ D = -1, -2;

Now the solution will be y = C₁e^-t + C₂e^-2t

Putting the values of x = e^t