MTH401 Final Term Solved Subjective Set 2

MTH401 FINAL TERM SOLVED PAPER · BY SHINING STAR · 1. For a differential equation

dy y x y identify P x Q x n dt x

There is a slight mistake in the question.

must be

1 ; dy dy

d d

P x Q x x t

n x x

2. For an Ist order linear differential equation

ln ln dy y x

dx x x x

,if the integrating

factor

x

is ln x then the solution of differential equation is

ln ln

ln

ln

is the required answer. ln x Q x dx C y x

x x dx C x y x

x dx C y x

x C y x

3. Find the annihilator operator of the function

and give reason for operator?

for 4cos 3sin

0 and =1,

Put in , we get

2(0) (0) (1)

for

It is a constant, with 1 and

put in D D n

n y x x

n

D D

D D

D

D

y

n

multiply both annihilator operator, we get

1 is the annihilator operator for 4cos 3sin D

D D x x

The above annihilator operator annihilates the given function. i.e. if we apply above

annihilator operator to the given function, result will come out to be zero.

4. What is an auxiliary equation of a homogeneous Differential equation? Also give its

two examples?

5. Find the auxiliary equation of the differential equation

" 2 ' te y y y t

When we solve non-homogeneous/homogeneous differential equations with constant coefficients

or variable coefficients, we start by assuming that a given particular function (which can never be

zero) is a solution of given differential equation. That function includes some particular variable

which in our case is m, What we do is we take derivatives of that function, put in given

differential equation and then take that factor out that particular function. Doing so, results in an

equation, which includes an algebraic equation. This algebraic equation can be equal to zero. It

will have roots, either complex or real roots. This equation is named as Auxiliary Equation.

However, characteristic equation is also another name for this equation.

Some examples of the auxiliary equation could be

For y’’+y = 0, auxiliary equation is m2+1=0

For y’’+2y’+y=0, auxiliary equation is m2+2m+1=0

Find the auxiliary equation of the differential equation

Auxiliary equation is m2-2m+1=0

6. What isWronskian of the function

cos sin cy c c x c x