linear independence of solutions
If you already have the solution e^x to a differential equation, then 2 e^x and -7 e^x tell you nothing new — they are just rescalings. To build a complete answer you need solutions that genuinely point in different directions, so that none of them is hiding inside the others. Linear independence is the precise statement of 'genuinely different'.
Solutions y_1, ..., y_n on an interval are linearly independent if the only constants making c_1 y_1 + ... + c_n y_n = 0 for all x in the interval are c_1 = ... = c_n = 0. If instead some nontrivial combination is identically zero, the functions are dependent and one of them is a combination of the rest. For solutions of a linear ODE there is a powerful shortcut: they are independent exactly when their Wronskian is nonzero somewhere on the interval — you need not test infinitely many combinations.
Why it matters: an n-th order linear homogeneous ODE has a solution space of dimension exactly n, so n linearly independent solutions form a basis and every solution is a unique combination of them. Independence guarantees you have enough genuinely distinct building blocks to satisfy any set of n initial conditions. Mistaking dependent solutions for independent ones leaves a deficient solution that cannot meet arbitrary initial data.
e^x and e^(2x) are linearly independent (no constant ratio between them), but e^x and 3e^x are dependent. The pair sin x, cos x is independent and forms a basis for solutions of y'' + y = 0.
For two functions, independence just means their ratio is not constant.
For arbitrary functions a vanishing Wronskian does not prove dependence; but for solutions of the SAME linear ODE on an interval, the Wronskian is either everywhere zero (dependent) or nowhere zero (independent) — that dichotomy is special to ODE solutions.