Question

1 Let f (t), g(t) be a continuous function on some interval I, and to e I. Prove that the initial value problem y'(t) f(t)y + g(t)y2, y(to) zo has a unique and continuous solution φ(t) on a small interval containing to, φ(t) satisfies the initial condition φ(to) = to.

Answer 1

Let $h(t,y):=f(t)y+g(t)y^2$ ; then we can write the given initial value problem as

u'(t)=h(t, y), y(to)=z0

Since
f(t),g(t)
are continuous, the function $h(t,y):=f(t)y+g(t)y^2$ is continuous, in both the variables . Moreover, this latter function is a polynomial in (with coefficient
f(t),g(t)
); we have

öh

which is also continuous in both the variables . Therefore, by existence-uniqueness theorem, there is an interval $I_0\subseteq I$ such that $t_0\in I_0$ and the given initial value problem has one and only one differentiable (and hence, continuous) solution
y = o(t)
on $I_0$ satisfying
o(to) = 20
. This is what is required.