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Magazine Problem 11
Find the general solution of the following equations. Express the solution explicitly as a function of the independent variable. $$x^{2} y^{\prime}(x)=y^{2}, x>0$$
Problem 11
Plot a direction field for the following differential equation with a graphing utility. Then find the solutions that are constant and determine which initial conditions \(y(0)=A\) lead to solutions that are increasing in time. $$y^{\prime}(t)=t(y-1), \quad 0 \leq t \leq 2,0 \leq y \leq 2$$
Problem 12
Make a sketch of the population function (as a function of time) that results from the following growth rate functions. Assume the population at time \(t=0\) begins at some positive value.
Problem 12
Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. \(A\) detailed direction field is not needed. $$y^{\prime}(t)=y-3, y(0)=1$$
Problem 12
Solve the following initial value problems. $$y^{\prime}(x)=-y+2, y(0)=-2$$
Problem 12
Verify that the given function \(y\) is a solution of the initial value problem that follows it. $$y(t)=8 t^{6}-3 ; t y^{\prime}(t)-6 y(t)=18, y(1)=5$$
Problem 13
Make a sketch of the population function (as a function of time) that results from the following growth rate functions. Assume the population at time \(t=0\) begins at some positive value.
Problem 13
Solve the following initial value problems. $$y^{\prime}(t)-2 y=8, y(0)=0$$
Problem 13
Use the window \([-2,2] \times[-2,2]\) to sketch a direction field for the following equations. Then sketch the solution curve that corresponds to the given initial condition. \(A\) detailed direction field is not needed. $$y^{\prime}(t)=4-y, y(0)=-1$$
Problem 13
Verify that the given function \(y\) is a solution of the initial value problem that follows it. $$y(t)=-3 \cos 3 t ; y^{\prime \prime}(t)+9 y(t)=0, y(0)=-3, y^{\prime}(0)=0$$
