Let’s consider the first order differential equation \(xy^{\,\prime }=x+y-\dfrac {y^2}{x}\) for \(x>0\).

(a)
Does this equation have any constant solutions?
(b)
This equation is not linear, not separable, and not Bernoulli. But it is nonlinear homogeneous. Let’s determine a differential equation \(u=\dfrac {y}{x}\) must satisfy.
(c)
Let’s solve this equation and then determine a general solution for \(y\).
(d)
Let’s determine the solution to the IVP \(xy^{\,\prime }=x+y-\dfrac {y^2}{x}\), \(y(1)=0\).

Now let’s consider the first order differential equation \(y^{\,\prime }=(x+y)^2\).

(a)
Does this equation have any constant solutions?
(b)
This equation is not linear, not separable, not Bernoulli, and not non-linear homogeneous. However, let’s determine a nice substitution to try in order to transform this differential equation into something nicer.
(c)
Let’s solve that equation, and then determine a general solution for \(y\).
(d)
What is the solution to the IVP \(y^{\,\prime }=(x+y)^2\), \(y(0)=0\)?
(e)
Let’s determine the interval \(I\) of maximum width on which this solution is valid.