Question

4. (D3 - 6D²D' + 110D 2 - 6D3) z = 0. find cf​

Answer 1

SOLUTION

TO DETERMINE

The complementary function

$\sf{({D }^{3} - 6 {D}^{2}D' + 11D {D'}^{2} - 6 {D'}^{3}) z = 0}$

EVALUATION

Here the given differential equation is

$\sf{({D }^{3} - 6 {D}^{2}D' + 11D {D'}^{2} - 6 {D'}^{3}) z = 0}$

Then the auxiliary equation is

$\sf{{m }^{3} - 6 {m}^{2}+ 11m - 6 = 0}$

$\sf{ \implies \: {m }^{3} - {m}^{2} - 5 {m}^{2}+5m + 6m - 6 = 0}$

$\sf{ \implies \: {m }^{2}(m - 1) - 5m(m - 1) + 6(m - 1)= 0}$

$\sf{ \implies \: (m - 1)({m }^{2}- 5m + 6)= 0}$

$\sf{ \implies \: (m - 1)(m - 2)(m - 3)= 0}$

$\sf{ \implies \: m = 1 \: , \: 2 \: ,\: 3}$

Hence the required complementary function is given by

$\sf{z = f_1(y + x) + f_2(y + 2x) + f_3(y +3 x)}$

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