MATLAB: Differential equation with mixed linear and log derivatives – proper setting

dependent variablesdifferential equationstrascendental equations

Hello everybody,
I'd like to solve for y = y(x) the following equation
that contains derivatives on both x and log(x).
When I input the equation as
syms y(x) f
eq = diff( log(y), log(x) ) + diff( log(diff(y,x)), log(x) ) + diff( y, log(x) )*log(x) + y ...
    == 1 + f;
I always get an error about the log in the differentiation
Second argument must be a variable or a nonnegative integer specifying the number of
differentiations.
I have tried to input it as a system of equations
syms y(x,z) f
eq1 = diff( log(y), x ) + diff( log(diff(y,z)), x ) + diff( y, x )*x + y ...
    == 1 + f;
eq2 = x == log(z);
But when I try to solve it
odes = [eq1;eq2];
sol = dsolve(odes);
I get an error that 
Symbolic ODEs must have exactly one independent variable.
I'm likely doing something wrong in managing the equations.
Can someone help me, please?
Thanks,
Patrizio

Best Answer

Using chain-rule, we can write
Therefore, the equation can be written as
syms y(x) f
eq = diff(log(y),x)*1/diff(log(x),x) + diff(log(diff(y,x)),x)*1/diff(log(x),x) + ...
    diff(y,x)*1/diff(log(x),x)*log(x) + y ...
    == 1 + f;
sol = dsolve(eq);
The symbolic solution is
>> sol
sol =
  ((2*C2*x^y + C2*f*x^y - C2*x^y*y + 2*C1*x^f*x^2)/(x^y*(f - y + 2)))^(1/2)
 -((2*C2*x^y + C2*f*x^y - C2*x^y*y + 2*C1*x^f*x^2)/(x^y*(f - y + 2)))^(1/2)
For numerical solution, try this
syms y(x) f
eq = diff(log(y),x)*1/diff(log(x),x) + diff(log(diff(y,x)),x)*1/diff(log(x),x) + ...
    diff(y,x)*1/diff(log(x),x)*log(x) + y ...
    == 1 + f;
eq2 = odeToVectorField(eq);
odeFun = matlabFunction(eq2, 'Vars', {'x', 'Y', 'f'});
xspan = [0.1 10];
xs = 0.1:0.001:10;
fv = rand(size(xs));
ffun = @(x) interp1(xs, fv, x);
ic = [1; 2];
[t, y] = ode45(@(x, y) odeFun(x, y, ffun(x)), xspan, ic);
plot(t, y);
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