Calculus How To

Linearly Independent Solutions: Simple Definition

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A linearly independent solution can’t be expressed as a linear combination of other solutions.

If f(x) and g(x) are nonzero solutions to an equation, they are linearly independent solutions if you can’t describe them in terms of each other. In math terms, we’d say that and is no c and k for which the expression

c f(x) + k g(x) = 0

is true.

Using a Graph to Find Linearly Independent Solutions

One way of determining whether a set of solutions is linearly independent is to graph them. If they are linearly independent, they will cross at exactly one place. Linear dependent solutions will either be parallel to each other or turn out to be actually the same line.

Suppose 4x + 2y = 6 and y = -x – 2 were your two solutions. Graphed, they become
linearly independent solutions

since they cross in just one place, they are linearly independent solutions.

The Wronskian Function

There is a special function called the Wronskian which can tell you whether two solutions (or functions of any sort) are linearly independent or not. The Wronskian W is defined as:

W (f, g)(t) = f(t) g'(t) – g(t) f'(t).


  • f and g are functions of t,
  • g'(t) is the first derivative of g,
  • f'(t) is the first derivative of f.

Suppose two functions f and g are differentiable on some interval I.

  • If W(f,g) (x) = 0 for all x in the interval I, the two functions are linearly dependent on I.
  • If W(f,g) (x) ≠ 0 for some x0 in the interval I, then the two functions are linearly independent.

As a simple example of this, let’s take a look at the two functions f(t) = 2 t2 and g(t) = t4. The Wronskian would be

f(t) g'(t) – g(t) f'(t) = 2 t2 · 4 t3 – t5 · 4t = 4t5.

Since this is not zero for some t (in fact, for all t except 0, the two functions are linearly independent.


Dawkins, Paul. More on the Wronskian. Paul’s Online Notes. Retrieved from on November 17, 2018Thomas, Weir, & Hass. Thomas’ Calculus, Chapter 17. Retrieved from on November 17, 2018.

Stephanie Glen. "Linearly Independent Solutions: Simple Definition" From Calculus for the rest of us!

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