Solve the following initial value problem. xy' = y + x sinx, y() = 0 [2...

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Solve the following initial value problem. xy' = y + x² sinx, y(π) = 0 [2 mark] Write down the characteristic equation of the following ODE and find the general solution. [2 mark] y"-y-6y=0 Write down the homogeneous ODE corresponding to the following differential equation and find the general homogeneous solution. Then find a particular solution by using the guess y(x) = ae-2x Finally, write down the general solution. [2 mark] y" +9y=e-2x Solve the following initial value problem. xy' = y + x² sinx, y(π) = 0 [2 mark] Write down the characteristic equation of the following ODE and find the general solution. [2 mark] y"-y-6y=0 Write down the homogeneous ODE corresponding to the following differential equation and find the general homogeneous solution. Then find a particular solution by using the guess y(x) = ae-2x Finally, write down the general solution. [2 mark] y" +9y=e-2x

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