(a)
Let solve the integro-differential IVP
\(\displaystyle y^{\,\prime }(t)+\int _0^t (t-x) y(x)\,dx=3t\), \(y(0)=0\).
(i)
First let’s re-write the integral as a convolution \(\displaystyle \left (f\ast g=\int _0^t f(x)g(t-x)\,dx\right )\).
(ii)
Now let’s take the Laplace transform of both sides, and solve for \(Y(s)=\mathcal {L}[y(t)]\).
(iii)
Now let’s find the inverse Laplace transform of \(Y(s)\).
(b)
Let’s consider the IVP \(y'=-2x+\sqrt {x+\frac {1}{y^2}}\), \(y(0)=1\). Here is an (approximate) direction field for this differential equation on \([-1,2]\times [0,3]\).
A direction field.
Figure 1: A direction field
(i)
Let’s use Euler’s method to approximate \(y(1)\) using \(5\) equal-sized steps.
(ii)
Let’s compare our estimate above with that which we can get by drawing.