Query 19 Differential Equations

Question:  Find the general solution of the equation

y '' + y = e^t sin(t).

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Question:  Find the general solution of the equation

y '' + y ' = 6 t^2

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Question:  Find the general solution of the equation

y '' + y ' = cos(t).

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Question:  Give the expected form of the particular solution to the given equation, but do not actually solve for the constants:

y '' - 2 y ' + 3 y = 2 e^-t cos(t) + t^2 + t e^(3 t)

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Question:  Give the expected form of the particular solution to the given equation, but do not actually solve for the constants:

y '' + 4 y = 2 sin(t) + cosh(t) + cosh^2(t).

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Question:  The equation

y '' + alpha y ' + beta y = t + sin(t)

has complementary solution y_C = c_1 cos(t) + c_2 sin(t) (i.e., this is the solution to the homogeneous equation).

Find alpha and beta, and solve the equation.

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Question:  Consider the equation

y '' - y = e^(`i * 2 t),

where `i = sqrt(-1). 

Using trial solution

y_P = A e^(i * 2 t)

find the value of A, which is in general a complex number (though in some cases the real or imaginary part of A might be zero)

Show that the real and imaginary parts of the resulting function y_P are, respectively, solutions to the real and imaginary parts of the original equation.

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