Example 2.2.1

Check solution to differential equation:

DSolve[y'[x]==x^2/y[x],y[x],x]

Example 2.2.4

Check solution to differential equation:

DSolve[y'[x]==(3x^2-1)/(3+2y[x]),y[x],x]

Example 2.2.24

Check solution to initial value problem:

DSolve[{y'[x]==(2-Exp[x])/(3+2y[x]),y[0]==-1},y[x],x]

Plot the solution, to check that the maximum occurs at Log[2]:

ContourPlot[{3 y+(y^2)==2 x-(E^x)+1,x==Log[2]},{x,-2,4},{y,-4,2},ContourStyle->{Red,Blue},ImageSize->{400,Automatic},AspectRatio->1]

Notice that we also get a minimum at x=Log[2].