Solve the differential equation by the method of variation of parameters, then solve it again by green function. Compare your answers. Are they the same?
[math]\frac{d^2y}{dx^2} + 4y = -e^x, 0 < x < l, y(0) = y(l) = 0[/math]
The homogenous solution is
[math]y(x) = c_1\cos 2x + c_2\sin 2x[/math]
Does the method of variation of parameters mean that I have to assume the constants are functions of x?
[math]c_1 = A(x)[/math][math]c_2 = B(x)[/math]
Then I can have two equations to solve simultaneously like this:
[math]A'(x)\cos 2x + B'(x)\sin 2x = 0[/math][math]-2A'(x)\sin 2x + 2B'(x)\cos 2x = -e^x[/math]
Is this always a valid choice?
[math]\frac{d^2y}{dx^2} + 4y = -e^x, 0 < x < l, y(0) = y(l) = 0[/math]
The homogenous solution is
[math]y(x) = c_1\cos 2x + c_2\sin 2x[/math]
Does the method of variation of parameters mean that I have to assume the constants are functions of x?
[math]c_1 = A(x)[/math][math]c_2 = B(x)[/math]
Then I can have two equations to solve simultaneously like this:
[math]A'(x)\cos 2x + B'(x)\sin 2x = 0[/math][math]-2A'(x)\sin 2x + 2B'(x)\cos 2x = -e^x[/math]
Is this always a valid choice?
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