Question

The differntial equation whose solution represent the family y=ae^3x+be^5x is y2 -8y1+lamday=0 then the valu e if lamda

Answer 1

SOLUTION

GIVEN

The differential equation whose solution represent the family

$\sf{y = a {e}^{3x} + b {e}^{5x} \: \: is \: \: y_2 - 8y_1 + \lambda y = 0 }$

TO DETERMINE

The value of $\sf{\lambda }$

EVALUATION

Here it is given that the differential equation has solution as

$\sf{y = a {e}^{3x} + b {e}^{5x} }$

So 3 and 5 are the zeroes of the auxiliary equation

So the auxiliary equation is

$\sf{(m - 3)(m - 5) = 0}$

$\sf{ \implies \: m(m - 5) - 3(m - 5) = 0}$

$\sf{ \implies \: {m}^{2} - 5m - 3m + 15 = 0}$

$\sf{ \implies \: {m}^{2} -8m + 15 = 0}$

So the differential equation is

$\sf{y_2 - 8y_1 + 15y = 0 }$

Now Comparing with the given differential equation

$\sf{y_2 - 8y_1 + \lambda y = 0 }$

We get

$\sf{ \lambda = 15 }$

FINAL ANSWER

$\boxed{ \: \: \: \sf{ \lambda = 15 \: } \: \: }$

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