Questions in Section Differential Equations: page 5

Strontium has a half-life of 28 days. After how many days a certain amount of Strontium-90 will be `1/3` of the initial amount?

Find the general solution of the linear second-order non-homogeneous differential equation

`(d^2y)/(dx^2)+2(dy)/(dx)+4y=4x-2`.

Transform the differential equation `(dy)/(dx)=2+2y+x+xy` to separable form by rearranging the right-hand side. Find:

  1. the general solution of the equation;
  2. the particular solution satisfying the initial condition `y(0)=1`.

Using any method you like find the general solution of the differential equation `(dy)/(dt)=ycost`.

Among the family of solutions find the one satisfying the initial condition `y(0)=3`.

Find the general solution of the differential equation `(d^2y)/(dx^2)-3(dy)/(dx)+2y=cosx`.

Find the general solution of the differential equation `(d^2y)/(dx^2)-3(dy)/(dx)-4y=cosx+x`.

This question relates to the matrix

`A=[(4,-2,-2),(0,1,0),(1,0,1)]`

  1. Show that `v=[[2],[0],[1]]` is an eigenvector of the matrix `A` and find its eigenvalue.
  2. Find the other eigenvalues of `A` and corresponding eigenvectors.
  3. Find `A^TA` and `A^T+A` and show that they are symmetric matrices.
  4. What is the general solution of the differential equation written in matrix-vector form as `y'=Ay`?

Given the differential equation `y''+2y'+10=x+sinx`, find the complementary function, `y_h`. Also write the form of the particular solution, `y_p`, that you would try. (You need not evaluate the coefficients.)

Consider the initial value problem `y'=-xy+2x`, `y(0)=5`.

  1. Show that the differential equation is separable. Also show it is linear.

    Find the general solution of the differential equation using either form and then find the solution to the initial value problem.

Find the general solution of the differential equation `(dy)/(dt)=e^-yt`.

Among the family of solutions find the one satisfying the initial condition `y(0)=2`.

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