%% Using dsolve to solve first order ODEs
%% Section 2.2 #9
% Solve $x^2y'=y \ln y - y'$.
%
% I solve it using *dsolve*. The equation has to be enclosed in single
% quotes and so does the independent variable. (If you leave out the
% argument for the independent variable, MATLAB assumes it is $t$, which
% isn't what you want here!)
syms x 
dsolve('x^2*Dy = y*log(y) - Dy','x')
%%
% You could also define the equation in MATLAB and give it a name, then use
% that in *dsolve*. 
eqn = 'x^2*Dy = y*log(y) - Dy'
dsolve(eqn,'x')
%% Solving an inital value problem
% Solve the same problem with $y(0)=2$.
%
% I add the initial value as the second argument. The independent variable
% has to be the last argument.
dsolve('x^2*Dy = y*log(y) - Dy','y(0)=2','x')
%%
% I could also combine the equation and initial value into one argument.
dsolve('x^2*Dy = y*log(y) - Dy, y(0)=2','x')
%% Solving with a general initial value $y(0)=c$
Y= dsolve('x^2*Dy = y*log(y) - Dy','y(0)=c','x')
%%
% This is messy. I'll simplify it.
Y = simplify(Y)
%%
% That helps a little.
%%
% You could define the initial value
% problem in MATLAB, give it a name and use it in *dsolve*.
IVP = 'x^2*Dy = y*log(y) - Dy, y(0)=c'
dsolve(IVP,'x')
%% Plotting the solutions for several initial values
% Plot the solutions with initial values 0,1,2,3,4
figure;
hold on
for j = 0:4
    ezplot(subs(Y,'c',j),[0 1])
end
hold off
axis([0 1 -1 25])
title('Solutions with initial value 0,...,4')
%%
% The examples in _Differential Equations with MATLAB_ use the *figure*
% command. At least here you can omit it.