Question

Consider the nonhomogeneous linear ODE with constant coefficients a and b y" + ay' + by = r(x). In class we found the particular integral yp(x) using the method of undetermined coefficients. This is very useful if r(x) if is not too complicated. Another method that can be used to find yp(x) is the method of variation of parameters. For this method let us first define the Wronskian of two functions. The Wronskian of the functions y1(x)and y2(x), denoted W(y1, y2) or W is defined to be W(y1, y2) = |y1 y2; y1' y2'| (i) Compute W(cos ????wx, sin wx). Now, if y1 and y2 are linearly independent solutions to the homogeneous equation y" + ay' + by = 0, then yp(x) = -y1 ?(y2r/W) dx + y2 ?(y1r/W) dx, where we ignore the constants of integration. (ii) Find out what the statement "y1 and y2 are linearly independent solutions" means and write down your understanding. Consider the ODE y" + y = sec x. (iii) Show that yh = A1 cos x + A2 sin x. (iv) Given that two linearly independent solutions are y1 = cos x and y2 = sin x, use the method of variation of parameters to find yp, and then write down the solution to the ODE.

          Consider the nonhomogeneous linear ODE with constant coefficients a and b

y" + ay' + by = r(x).

In class we found the particular integral yp(x) using the method of undetermined coefficients. This is very useful if r(x) if is not too complicated. Another method that can be used to find yp(x) is the method of variation of parameters. For this method let us first define the Wronskian of two functions.
The Wronskian of the functions y1(x)and y2(x), denoted W(y1, y2) or W is defined to be

W(y1, y2) = |y1 y2; y1' y2'|

(i) Compute W(cos ????wx, sin wx).

Now, if y1 and y2 are linearly independent solutions to the homogeneous equation

y" + ay' + by = 0,

then

yp(x) = -y1 ?(y2r/W) dx + y2 ?(y1r/W) dx,

where we ignore the constants of integration.

(ii) Find out what the statement "y1 and y2 are linearly independent solutions" means and write down your understanding.

Consider the ODE

y" + y = sec x.

(iii) Show that yh = A1 cos x + A2 sin x.

(iv) Given that two linearly independent solutions are y1 = cos x and y2 = sin x, use the method of variation of parameters to find yp, and then write down the solution to the ODE.

Consider the nonhomogeneous linear ODE with constant coefficients a and b

y" + ay' + by = r(x).

In class we found the particular integral yp(x) using the method of undetermined coefficients. This is very useful if r(x) if is not too complicated. Another method that can be used to find yp(x) is the method of variation of parameters. For this method let us first define the Wronskian of two functions.
The Wronskian of the functions y1(x)and y2(x), denoted W(y1, y2) or W is defined to be

W(y1, y2) = |y1 y2; y1' y2'|

(i) Compute W(cos ????wx, sin wx).

Now, if y1 and y2 are linearly independent solutions to the homogeneous equation

y" + ay' + by = 0,

then

yp(x) = -y1 ?(y2r/W) dx + y2 ?(y1r/W) dx,

where we ignore the constants of integration.

(ii) Find out what the statement "y1 and y2 are linearly independent solutions" means and write down your understanding.

Consider the ODE

y" + y = sec x.

(iii) Show that yh = A1 cos x + A2 sin x.

(iv) Given that two linearly independent solutions are y1 = cos x and y2 = sin x, use the method of variation of parameters to find yp, and then write down the solution to the ODE.

Added by Isabel W.

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