Chapter 3

Solve the generic equation \(y^{\prime}=x+a x y .\) How does varying \(a\) change the behavior?

Solve \(y^{\prime}-y=e^{k t}\) with the initial condition \(y(0)=0 .\) As \(k\) approaches 1, what happens to your formula?

Solve the following initial-value problems starting from \(y_{0}=10 .\) At what time does \(y\) increase to 100 or drop to \(1 ?$$\frac{d y}{d t}=e^{-4 t}\)

The differential equation \(y^{\prime}=3 x^{2} y-\cos (x) y^{\prime \prime}\) is linear.

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from \(y(t=0)=-10\) to \(y(t=0)=10\) increasing by \(2 .\) Is there some critical point where the behavior of the solution begins to change?[I] \(y^{\prime}=y(x)\)

The differential equation \(y^{\prime}=x-y\) is separable.

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from \(y(t=0)=-10\) to \(y(t=0)=10\) increasing by \(2 .\) Is there some critical point where the behavior of the solution begins to change?[T] \(x y^{\prime}=y\)

Recall that a family of solutions includes solutions to a differential equation that differ by a constant. For the following problems, use your calculator to graph a family of solutions to the given differential equation. Use initial conditions from \(y(t=0)=-10\) to \(y(t=0)=10\) increasing by \(2 .\) Is there some critical point where the behavior of the solution begins to change?\([\mathrm{T}] y^{\prime}=t^{3}\)

Find the general solution to the differential equations. $$ y^{\prime}=x^{2}+3 e^{x}-2 x $$

Find the general solution to the differential equations. $$ y^{\prime}=2^{x}+\cos ^{-1} x $$