Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1994 • Question 2

Question

Math 6410 • ODE Fall 1994 • Question 2

Consider the initial value problem \[ x'=f(t,x),\qquad x(t_0)=x_0 \] on the set \(X:\lvert t-t_0\rvert \le a\), \(\lvert x-x_0\rvert \le b\). Let \(M=\max \lvert f(t,x)\rvert \) on \(X\), \(h=\min (a,b/M)\). Assume that \(f\) is continuous on \(X\), \(L\ge 0\) and \(f\) satisfies the Lipschitz condition \(\lvert f(t,x)-f(t,y)\rvert \le L\lvert x-y\rvert \) for \((t,x)\in X\), \((t,y)\in X\).

Prove the Picard–Lindelöf existence and uniqueness theorem: There exists a unique solution \(x(t)\) defined on the interval \(\lvert t-t_0\rvert \le h\).

Consider the initial value problem
\[
x'=f(t,x),\qquad x(t_0)=x_0
\]
on the set \(X:\lvert t-t_0\rvert\le a\), \(\lvert x-x_0\rvert\le b\). Let
\(M=\max\lvert f(t,x)\rvert\) on \(X\), \(h=\min(a,b/M)\). Assume that \(f\) is continuous
on \(X\), \(L\ge0\) and \(f\) satisfies the Lipschitz condition
\(\lvert f(t,x)-f(t,y)\rvert\le L\lvert x-y\rvert\) for \((t,x)\in X\), \((t,y)\in X\).

Prove the Picard--Lindel\"of existence and uniqueness theorem: There exists a
unique solution \(x(t)\) defined on the interval \(\lvert t-t_0\rvert\le h\).

Topic Tags

Existence and Uniqueness Theorems Global Existence and Finite-Time Blow-Up Gronwall Inequality and Continuous Dependence Poincare Maps and Periodic Solutions