fanyattehedzg

2021-12-30

Find the solution of the following Second Order Differential Equation

Mason Hall

Beginner2021-12-31Added 36 answers

We have given

$y9y=0$

Auxiliary equation of given differential equation

${m}^{2}-9=0$

$(m-3)(m+3)=0$

$m=3,-3$

Therefore, general solution of differential equation

$y={c}_{1}{e}^{3x}+{c}_{2}{e}^{-3x}$

We have$y\left(0\right)=2$

$\Rightarrow y\left(0\right)={c}_{1}{e}^{3\left(0\right)}+{c}_{2}{e}^{-3\left(0\right)}$

$2={c}_{1}+{c}_{2}$ (i)

$y}^{\prime}\left(x\right)=3{c}_{1}{e}^{3x}-3{c}_{2}{e}^{-3x$

We have${y}^{\prime}\left(0\right)=0$

$\Rightarrow {y}^{\prime}\left(x\right)=3{c}_{1}{e}^{3\left(0\right)}-3{c}_{2}{e}^{-3\left(0\right)}$

$0=3{c}_{1}-3{c}_{2}$

$0={c}_{1}-{c}_{2}$ (2)

Now, solve equation (1) and (2) we get

${c}_{1}=1,{c}_{2}=1$

Therefore, particular solution is

$y={e}^{3x}+{e}^{-3x}$

Auxiliary equation of given differential equation

Therefore, general solution of differential equation

We have

We have

Now, solve equation (1) and (2) we get

Therefore, particular solution is

Linda Birchfield

Beginner2022-01-01Added 39 answers

The auxiliary equation for this equation is:

On solving this equation for "m".

Then, the solution of this given differential equation is:

Given that,

So, replace x with 0 and y with 2 in it.

Now, differentiate the obtained solution with respect to x.

And, also given that,

So, replace x with 0 and y with 0 in it.

Now, solve equations (1) and (2) by the elimination or substitution method.

From equation (2),

Then,

Thus, the solution of the given differential equation is:

karton

Expert2022-01-09Added 613 answers

22+64

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