UNIT-5

Partial differential equation: Higher order

Question-1: Solve

Solution. Given equation can be written as

Here auxiliary equation is

Hence the required solution is

Question-2: Solve.

Solution. Given equation can be written as

A.E. Is

Hence the required solution is

Question-3: Solve

Solution. Here the auxiliary equation is

Its root are

The complementary function is

On putting y = 2 and x = 0 in (1) we get

2 =A

On putting A = 2 in (1) we have

On Differentiating (2) we get

But,

On putting x = 0, we get

(2) becomes,

Question-4: Using the method of separation of variables, solve

Solution. 

Let, u = X(x). T (t).                     (2)

Where X is a function of x only and T is a function of t only.

Putting the value of u in (1), we get

(a) 

On integration log X = cx + log a = log

(b)

On integration

Putting the value of X and T in (2) we have

But,

i.e.

Putting the value of a b and c in (3) we have

Which is the required solution.

Question-5: Use the method of separation of variables to solve equation

Given that v = 0 when t→ as well as v =0 at x = 0 and x = 1.

Solution.

Let us assume that v = XT where X is a function of x only and T that of t only

Substituting these values in (1), we get

Let each side of (2) equal to a constant

Solving (3) and (4) we have

Putting x = 0, v = 0 in (5) we get

On putting the value of in (5) we get

Again putting x = l, v= 0 in (6) we get

Since cannot be zero.

Inputting the value of p in (6) it becomes

Hence,

This equation satisfies the given condition for all integral values of n. Hence taking n = 1, 2, 3,… the most general solution is

Question-6: A rod of length 1 with insulated sides is initially at a uniform temperature u. Its ends are suddenly cooled to 0° Celsius and are kept at that temperature. Prove that the temperature function u (x, t) is given by

Where is determined from the equation.

Solution. Let the equation for the conduction of heat be

Let us assume that u = XT, where X is a function of x alone and T that of t alone.

Substituting these values in (1) we get

i.e.

Let each side be equal to a constant

And

Solving (3) and (4) we have

Putting x = 0, u = 0 in (5), we get

(5) becomes

Again putting x = l, u =0 in (6), we get

Hence (6) becomes

This equation satisfies the given conditions for all integral values n. Hence taking n = 1, 2, 3,…, the most general solution is

By initial conditions

Question-7: The ends A and B of a rod 20 cm long having the temperature at 30 degree Celsius and at 80 degree Celsius until steady state prevails. The temperature of the ends are changed to 40 degree Celsius and 60 degree Celsius respectively. Find the temperature distribution in the rod at time t.

Solution. The initial temperature distribution in the rod is

And the final distribution (i.e. steady state) is

To get u in the intermediate period, reckoning time from the instant when the end temperature were changed we assumed

Where is the steady state temperature distribution in the rod (i.e. temperature after a sufficiently long time) and is the transient temperature distribution which tends to zero as t increases.

Thus,

Now satisfies the one dimensional heat flow equation

Hence u is of the form

Since

Hence

Using the initial condition i.e.

Putting this value of n (1), we get

Question-8: Find the deflection of a vibrating string of unit length having fixed ends with initial velocity zero and initial deflection f (x)=k (sinx –sin2x)

Solution. By d’Alembert’s method, the solution is

i.e., the given boundary corrections are satisfied.