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Verify the solution of the differential equation. NOTE: Observe a structure of the solution, then...

Question:

Verify the solution of the differential equation.

NOTE: Observe a structure of the solution, then use appropriate rules.

{eq}y" - 5y' + 4y = 0 {/eq}

Solution {eq}y = e^{4x} {/eq}

Differential Equation:

The equation can be found by replacing y by operator D and then finding the roots of the new equation. We will get the two roots. Then from the roots, we can write the general equation involving two constants.

Answer and Explanation: 1

To solve the differential equation we will replace the double derivative of y by {eq}D^{2} {/eq}, the single derivative of y by D and then put it equal to 0.

Let us now find the roots of the equation

{eq}y''-5y'+4y=0\\ D^{2}-5D+4=0\\ D=1,4 {/eq}

The roots of the equation are 1,4

Now writing the general equation we get

{eq}y=c_{1}e^{x}+c_{2}e^{4x} {/eq}


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