Q7Advanced Engineering Mathematics

Question

Solve linear difference equation u_{n + 2} + 6 u_{n + 1} + 9 u_{n} = 2^{n} given u_{0} = 1, u_{1} = 1.

Answer 1

The problem requires solving the second-order, linear, constant-coefficient, non-homogeneous difference equation $u_{n + 2} + 6 u_{n + 1} + 9 u_{n} = 2^{n}$ . The total comprehensive solution is mathematically constructed by independently finding two distinct components: the Complementary Function (CF) which represents the natural unforced response of the system, and the Particular Integral (PI) which represents the forced response to the specific input $2^{n}$ . The final solution is $u_{n} = C F + P I$ .

1. Determining the Complementary Function

To find the CF, we set the right-hand side forcing function exactly to zero, creating a homogeneous equation. We then utilize the mathematical shift operator $E$ (where $E u_{n} = u_{n + 1}$ ) to rewrite the equation algebraically: $(E^{2} + 6 E + 9) u_{n} = 0$ .

This yields the quadratic characteristic auxiliary equation $E^{2} + 6 E + 9 = 0$ . Factoring this perfect square gives $(E + 3)^{2} = 0$ . This indicates real, repeated roots at $E_{1} = E_{2} = - 3$ . According to the fundamental theorems of difference equations, the structural form of the CF for repeated roots is:

u_{n}^{(c)} = (C_{1} + C_{2} n) (- 3)^{n}

2. Determining the Particular Integral

The PI addresses the specific non-homogeneous forcing function $f (n) = 2^{n}$ . We apply inverse operator methods to find it. The PI is mathematically defined as $\frac{1}{F ( E )} f (n)$ .

P I = \frac{1}{E ^{2} + 6 E + 9} (2^{n}) = \frac{1}{( E + 3 ) ^{2}} (2^{n})

When the forcing function is an exponential of the form $a^{n}$ , the inverse operator rule allows us to directly substitute $E$ with the constant base $a$ , provided the resulting denominator does not equal zero. Here, $a = 2$ .

P I = \frac{1}{( 2 + 3 ) ^{2}} (2^{n}) = \frac{1}{5 ^{2}} (2^{n}) = \frac{1}{25} 2^{n}

3. Constructing the General Solution

Combining both distinct components, the general overall solution to the difference equation is:

u_{n} = (C_{1} + C_{2} n) (- 3)^{n} + \frac{1}{25} 2^{n}

The problem provides specific initial boundary conditions $u_{0} = 1$ and $u_{1} = 1$ . We rigorously substitute these into our general solution to uniquely determine the unknown constants $C_{1}$ and $C_{2}$ .

At $n = 0$ : $u_{0} = (C_{1} + 0) (- 3)^{0} + \frac{1}{25} (1) = C_{1} + \frac{1}{25} = 1 ⟹ C_{1} = \frac{24}{25}$ .

At $n = 1$ : $u_{1} = (C_{1} + C_{2}) (- 3)^{1} + \frac{1}{25} (2) = - 3 (\frac{24}{25} + C_{2}) + \frac{2}{25} = 1$ .

- \frac{72}{25} - 3 C_{2} + \frac{2}{25} = 1 ⟹ - 3 C_{2} - \frac{70}{25} = 1 ⟹ - 3 C_{2} = 1 + \frac{70}{25} = \frac{95}{25}

C_{2} = - \frac{95}{75} = - \frac{19}{15}

Substituting the exact calculated constants back yields the final, definitively proven specific solution sequence:

u_{n} = (\frac{24}{25} - \frac{19}{15} n) (- 3)^{n} + \frac{1}{25} 2^{n}

Q7Advanced Engineering Mathematics

Question

Answer

1. Determining the Complementary Function

2. Determining the Particular Integral

3. Constructing the General Solution