Verify that the function y=e^{-3x} is a solution to the

Roger Smith

Roger Smith

Answered question

2021-12-18

Verify that the function y=e3x is a solution to the differential equation d2ydx2+dydx6y=0?

Answer & Explanation

Becky Harrison

Becky Harrison

Beginner2021-12-19Added 40 answers

Step 1 
To validate the function y=e3x is a solution of the differential equations d2y dx 2+ dy  dx 6y=0
That is the function y=e3x satisfies the equation d2y dx 2+ dy  dx 6y=0 ...(1) 
Step 2 
So,  dy  dx =d dx (e3x)=e3x(3)=3e3x 
d2y dx 2=d dx 3(e3x)=(3)d dx (e3x)=3(e3x(3))=9e3x 
6y=6e3x 
Hence, d2y dx 2+ dy  dx 6y 
=9e3x3e3x6e3x 
=9e3x9e3x 
=0 
As a result, equation (1) is true.
The given differential equation has a solution in the form of the given function, y=e3x 

Orlando Paz

Orlando Paz

Beginner2021-12-20Added 42 answers

Given, y=e3x
On differentiating with x, we get
dydx=3e3x
On differentiating again with x, we get
d2ydx2=9e3x
Now let's see what is the value of d2ydx2+dydx6y
=9e3x3e3x6e3x
=0
Conclusion: Therefore, y=e3x is the solution of d2ydx2+dydx6y=0
RizerMix

RizerMix

Expert2021-12-29Added 656 answers

y=e3x
x-based differentiation
dydx=3e3x
Again differentiating, d2ydx2=9e3x
Substituting in d2ydx2+dydx6y
9e3x3e3x(6e3x)=0
Hence y=e3x is a solution.

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