Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval $I$ of definition for each solution. $$y^{\prime \prime}-6 y^{\prime}+13 y=0 ; \quad y=e^{3 x} \cos 2 x$$

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Ekaveera Kumar

University College of Engineering Osmania University

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Verify that the indicated function is an explicit solution of the given differential equation. Assume an appropriate interval $I$ of definition for each solution. $$y^{\prime \prime}-6 y^{\prime}+13 y=0 ; \quad y=e^{3 x} \cos 2 x$$

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We need to verify whether the differential equation of y double dash minus 6 y dash plus 13 y equal to 0 is satisfied by the function. The power 3 x cos 2 x. So, let's find the derivative y dash is equal to. Let'S use the product rule power 3 x into derivative of 2 x, which is 2 x, plus cos 2 x into derivative power 3 x, which is 3 power 3 x. So, basically, by bold dash is the power 3 x into 3 cos 2 x minus 2 sine 2 y dash. Let'S differentiate for the y double dash is to the part the into derivative bracket, which is minus 6 sine 2 x. Minus 4 cos 2 x, plus brackett 3 cos 2 x minus 2 sine 2 x into derivative. He paweth, which is 3 parthso. That will be minus 6, a power 3 x sine 2 x minus 4. A power 3 x cos 2 x plus 9. A power 3 x cos 2 x minus 6 power 3 x sine 2 x petite. Like terms it will be 5 power, 3 x cos 2 x, minus 12 power 3 x sine 2 x. So this is going by double dash. Multiply this with negative 6. What is negative 6 y dash negative 6 y dash is negative. 18. The power 3 x cos 2 x, plus 12 power 3 x, 2 x. I multiplied with this with negative 6 point. That'S why negative 6 y dash and also what is 13, why? Why is this? What is 13 y 13 a power 3 x cos 2 x. It is 13 power, 3 x, x, 2 x, to add a this cora. This corso here is y double dash. 6 y dash is 131 point, but if you are here, 5 power 3 x, cos 2 x, plus 13 power 3 x. Cos 218 is canceled. With this and the star were getting cancel, that's it.

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