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Consider the differential equation y'' − y' − 30y = 0. Verify that the functions e−5x and e6x form a fundamental set of solutions of the differential equation on the interval (−[infinity], [infinity]). The functions satisfy the differential equation and are linearly independent since the Wronskian W (e−5x, e6x) = ___________?

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mavila18

Answer:

Step-by-step explanation:

We have to take the derivatives for both functions and replace in the differential equation. Hence

for y=e^{-5x}:

[tex]y(x)=e^{-5x}\\y'(x)=-5e^{-5x}\\y''(x)=25e^{-5x}\\[/tex]

for y=e^{6x}:

[tex]y(x)=e^{6x}\\y'(x)=6e^{6x}\\y''(x)=36e^{6x}\\[/tex]

Now we replace in the differential equation  y'' − y' − 30y = 0

for y=e^{-5x}:

[tex]25e^{-5x}+5e^{-5x}-30e^{-5x}=0\\25+5-30=0[/tex]

for y=e^{6x}:

[tex]36e^{6x}-6e^{6x}-30=0\\36-6+30=0[/tex]

Now, to know if both function are linearly independent we calculate the Wronskian

[tex]W(f,g)=fg'-f'g[/tex]

[tex]W(e^{-5x},e^{6x})=(e^{-5x})(6e^{6x})-(-5e^{-5x})(e^{6x})\neq 0[/tex]

I hope this is useful for you

Best regard

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