september 19
 Sketch the vector fields for each value of the parameter; indicate equilibria and their stability:
 \[
x’=3\sin(x)+x+a, a=3,0,3;
\]
 \[
x’=\tan(x)3x+a, a=5,0,5;
\]
 \[
x’=e^xx+a, a=1,0,1;
\]
 Find the three consecutive Picard iterations for the equation.
Solve the differential equation; compare the Taylor expansion of the result with the third Picard iteration.
 \[
y’=y+1, y(0)=0.
\]
 \[
y’=y^2+1, y(0)=0.
\]
 \[
y’=y(1y), y(0)=1/2.
\]

Solve:
 \[
xy”’=y”xy”;
\]
 \[
yy”+y=(y’)^2;
\]
 \[
yy”’+3y’y”=0.
\]
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