Question

d
5
For the differential equation (D-1)4y = e 2x where D =
the complimentary function is
a) (C1x3 + c2x2 + C3x + C4) e 2x
b)CeX + Cze-* + C3 COS X + C4 sin x
c)(C123 + C2x2 + C3x + C4)e*
d) Cie* + cze-x + c3 cos 2x + ca sin 2x​

Answer 1

Answer:

Here, Option (a) is correct.

Answer 2

Concept:

We need to recall the concept of differential equations to solve this question.

• Differential Equation (D.E.) is defined as the equation  which involves the dependent variable, independent variable and the derivative of dependent variable with respect to independent variable.
• Complementary function is formed by the roots of the characteristic equation of D.E.

Given:

The differential equation is $(D-1)^{4}$ =  $e^{2x}$.

To find:

The Complementary function of the given differential equation.

Solution:

The auxiliary equation or characteristics equation of given D .E. is given by :          $(m-1)^{4} = 0$

(m-1)(m-1)(m-1)(m-1) = 0

                          m = 1,1 ,1,1

So, the given D.E. has 4 repeated roots given by m=1.

Hence, the complementary function (C.F.) is:

y(x) = $(C_{1}x^{3} +C_{2}x^{2} +C_{3}x+C_{4}) e^{x}$