Question

1. Let C ? R, x? ? Domain(y), and k ? Range(y). Construct an example of a second order, ordinary differential equation and verify a solution using the following steps. (a) Choose a function y(x) that is twice differentiable. Compute y'(x) and y''(x). (b) Choose functions (they may be constant) a?(x) and a?(x). Then compute f(x) = a?(x)y''(x) + a?(x)y'(x) (c) Show that y(x) + C is a solution of f(x) = a?(x)y''(x) + a?(x)y'(x). (d) For what value(s) of C does y(x?) = k? Choose x? and k to give an example of an initial value problem using the ODE constructed in part (b). Provide the solution of the IVP. (e) Use a similar method (or share your own method) to construct another second-order IVP that you can verify the solution of without using any ODE techniques other than the FTOC. Make sure to include justification for the solution and verify it.

          1. Let C ? R, x? ? Domain(y), and k ? Range(y). Construct an example of a second order,
ordinary differential equation and verify a solution using the following steps.
(a) Choose a function y(x) that is twice differentiable. Compute y'(x) and y''(x).
(b) Choose functions (they may be constant) a?(x) and a?(x). Then compute
f(x) = a?(x)y''(x) + a?(x)y'(x)
(c) Show that y(x) + C is a solution of f(x) = a?(x)y''(x) + a?(x)y'(x).
(d) For what value(s) of C does y(x?) = k? Choose x? and k to give an example of an initial
value problem using the ODE constructed in part (b). Provide the solution of the IVP.
(e) Use a similar method (or share your own method) to construct another second-order IVP
that you can verify the solution of without using any ODE techniques other than the
FTOC. Make sure to include justification for the solution and verify it.

1. Let C ? R, x? ? Domain(y), and k ? Range(y). Construct an example of a second order,
ordinary differential equation and verify a solution using the following steps.
(a) Choose a function y(x) that is twice differentiable. Compute y'(x) and y”(x).
(b) Choose functions (they may be constant) a?(x) and a?(x). Then compute
f(x) = a?(x)y”(x) + a?(x)y'(x)
(c) Show that y(x) + C is a solution of f(x) = a?(x)y”(x) + a?(x)y'(x).
(d) For what value(s) of C does y(x?) = k? Choose x? and k to give an example of an initial
value problem using the ODE constructed in part (b). Provide the solution of the IVP.
(e) Use a similar method (or share your own method) to construct another second-order IVP
that you can verify the solution of without using any ODE techniques other than the
FTOC. Make sure to include justification for the solution and verify it.

Added by Jessica R.

Question

Please give Ace some feedback