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$

A.(Ax + B){e^x} \\

B.A\cos x + B\sin x \\

C.(Ax + B){e^{2x}} \\

D.(Ax + B){e^{ - x}} \\

$

Answer

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Hint: Use auxiliary equation concept and ${y_c} = {e^{ax}}(A\cos \beta x + B\sin \beta x)$to find the complementary function of $({D^2} + 1)y = {e^{2x}}$.

Auxiliary equation is an equation with one variable and equated to zero, which is derived from a given linear differential equation and in which the coefficient and power of the variable in each term correspond to the coefficient and order of a derivative in the original equation.

Complete step-by-step answer:

Hence, the auxiliary equation of above differential equation is $({D^2} + 1)y = 0$

For complementary function let $D = m$

Hence, $ \Rightarrow ({m^2} + 1)y = 0$

OR

$

\Rightarrow {m^2} + 1 = 0 \\

\Rightarrow {m^2} = - 1 \\

$

$ \Rightarrow $$m = \sqrt { - 1} = \pm i$

Since the roots are complex , so by formula ${y_c} = {e^{ax}}(A\cos \beta x + B\sin \beta x)$

Where , $a = 0$and $\beta = 1$

Hence , by substituting the values in the formula we get

${y_c} = (A\cos x + B\cos x)$

Note: It is advisable to remember various characteristic roots and their formulas to save time. Eventually it will be difficult to mug up every formula but with practice things get easier.

Auxiliary equation is an equation with one variable and equated to zero, which is derived from a given linear differential equation and in which the coefficient and power of the variable in each term correspond to the coefficient and order of a derivative in the original equation.

Complete step-by-step answer:

Hence, the auxiliary equation of above differential equation is $({D^2} + 1)y = 0$

For complementary function let $D = m$

Hence, $ \Rightarrow ({m^2} + 1)y = 0$

OR

$

\Rightarrow {m^2} + 1 = 0 \\

\Rightarrow {m^2} = - 1 \\

$

$ \Rightarrow $$m = \sqrt { - 1} = \pm i$

Since the roots are complex , so by formula ${y_c} = {e^{ax}}(A\cos \beta x + B\sin \beta x)$

Where , $a = 0$and $\beta = 1$

Hence , by substituting the values in the formula we get

${y_c} = (A\cos x + B\cos x)$

Note: It is advisable to remember various characteristic roots and their formulas to save time. Eventually it will be difficult to mug up every formula but with practice things get easier.

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