Differentiating results in us knowing how large y-output changes compared to qan amount of x-input in the function f(x)=y.
Integrating is, uh, just “the summation” of all y-outputs as a result of x-inputs.
Now, what does a DE tell me? And what does solving it do?
In: Mathematics
Other comments have already described a lot so maybe my contribution can be the introductory example (assuming knowledge of calculus): f(x) = f'(x). This is a DE, and it can be solved by dividing both sides by f(x) to get f'(x)/f(x) = 1 and then integrating both sides with respect to x. On the right this just becomes x + C_1, and on the left we can use u-substitution by letting u = f(x) so that du = f'(x)dx, meaning the integral becomes the integral of 1/u with respect to u. This is ln(u) + C_2 (or ln(|x|) + C_2 depending on how rigorous we’re being). So we can substitute f(x) back in to get ln(f(x)) + C_2 = x + C_1. Then solve for f(x):
ln(f(x)) + C_2 = x + C_1
exp(ln(f(x)) + C_2) = exp(x + C_1)
exp(ln(f(x)))exp(C_2) = exp(x)exp(C_1)
exp(ln(f(x))) = exp(C_1)/exp(C_2) * exp(x)
f(x) = Ce^x (where we usually just write C = e^C_1 / e^C_2 )
And there we have it, we have solved the differential equation where a function’s derivative equals the original function, to find out what functions satisfy such an equation.
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