Compute the Wronskian determinant of two or more functions instantly. Test linear independence for differential equations with full, LaTeX-rendered step-by-step working.
Built for students, engineers and researchers who need reliable Wronskian computations.
Automatically differentiates each function using a real computer-algebra engine — no lookup tables or approximations.
Solve 2×2, 3×3 and 4×4 Wronskian matrices for higher-order linear differential equations.
Every matrix, derivative and determinant expansion is beautifully typeset with KaTeX.
Evaluate the Wronskian symbolically or at a specific numeric point instantly.
Instantly see whether your functions are linearly independent based on the computed result.
Copy the full solution or download it as a text file for assignments and reports.
Comfortable calculating any time of day with a persistent theme toggle.
Works cleanly on desktop, tablet, and mobile with zero horizontal scroll.
Instant feedback on malformed expressions before you ever hit calculate.
Type in 2, 3 or 4 functions of a single variable — polynomials, trig, exponential, or combinations.
The tool arranges the functions and their successive derivatives into a square matrix automatically.
A real determinant expansion (cofactor method) is applied and simplified symbolically.
See the Wronskian result and whether your functions are linearly independent, with full working shown.
Anyone working through a course on ordinary differential equations eventually runs into the Wronskian, and honestly, it looks more intimidating on paper than it actually is in practice. At its core, the Wronskian is just a determinant built from a set of functions and their derivatives. Its entire job is to answer one simple question: are these functions linearly independent, or is one secretly just a scaled combination of the others? That single piece of information turns out to be enormously useful when you are trying to decide whether a collection of solutions to a differential equation actually forms a complete, usable "fundamental set" of solutions.
For two functions f(x) and g(x), the Wronskian is written as W(f, g) = f·g′ − f′·g. You are simply multiplying each function by the derivative of the other and subtracting. For three functions, the pattern extends into a full 3×3 determinant: the first row holds the functions themselves, the second row holds their first derivatives, and the third row holds their second derivatives. This is exactly the structure our calculator builds automatically the moment you type in your expressions, and it scales the same way for four functions with third derivatives added as a fourth row.
The Wronskian matrix is nothing more than an organized table. Reading left to right across the top row gives you the original functions. Every row underneath is simply the derivative of the row above it. Once that matrix is assembled, calculating the Wronskian is just a matter of taking its determinant — the same determinant technique you would use on any square matrix, whether that is a simple 2×2 shortcut or full cofactor expansion for larger sizes.
When you are solving a second-order linear homogeneous differential equation, you typically find two independent solutions and combine them into a general solution using arbitrary constants. But how do you know your two solutions are actually independent, rather than one being a disguised multiple of the other? That is precisely where the Wronskian method comes in. If you compute the Wronskian of your candidate solutions and it turns out nonzero at any single point in the interval you care about, Abel's theorem guarantees it stays nonzero throughout that interval, confirming independence for the whole domain.
The underlying theorem is elegant: if a set of functions is linearly dependent, their Wronskian is identically zero everywhere. The reverse implication needs a little more care — a zero Wronskian does not always guarantee dependence for arbitrary functions, but for solutions of a common linear differential equation, it does. This distinction trips up a lot of students, so our calculator explicitly flags whether your functions look independent based on the computed value, while reminding you of the underlying nuance.
Take f(x) = sin(x) and g(x) = cos(x). Their derivatives are cos(x) and −sin(x) respectively, so the Wronskian becomes sin(x)·(−sin(x)) − cos(x)·cos(x), which simplifies neatly to −1 using the Pythagorean identity. Since that value never touches zero, sine and cosine are linearly independent everywhere — exactly the kind of confirmation you would want before building a general solution around them. Try polynomial examples like x and x², or exponential pairs like eˣ and e²ˣ, directly in the calculator above to see the full working.
| Functions | Wronskian | Independent? |
|---|---|---|
| x, x² | x² | Yes (except at x = 0) |
| sin(x), cos(x) | −1 | Yes |
| eˣ, e²ˣ | e³ˣ | Yes |
| x, 2x | 0 | No |
Using the tool is meant to feel effortless. Pick how many functions you want to test, type each expression using standard notation such as sin(x), x^2 or e^x, and optionally provide a numeric point if you want a concrete number rather than a symbolic expression. Press calculate, and the tool scrolls straight to a fully worked solution: the constructed matrix, every derivative taken along the way, the determinant expansion, and a final plain-English verdict on linear independence. You can then copy the entire solution to your clipboard or download it as a text file for your notes or assignment.
Beyond textbook exercises, the Wronskian shows up whenever engineers or physicists need to confirm that a fundamental set of solutions genuinely spans the full solution space of a system — think mechanical vibration models, RLC circuit analysis, or coupled differential equations in control theory. Being able to check independence quickly, without manually grinding through derivatives and determinants, saves real time when you are validating a model rather than learning the theory for the first time.
You arrange your functions and their successive derivatives into rows of a square matrix, then take the determinant of that matrix. For two functions f and g, this simplifies to f·g′ − f′·g.
A nonzero Wronskian at any point in an interval proves the functions are linearly independent on that interval. A Wronskian that is identically zero suggests possible dependence.
For three functions, build a 3×3 matrix with the functions in row one, their first derivatives in row two, and their second derivatives in row three, then compute its determinant.
It confirms whether a set of solutions to a linear homogeneous differential equation is independent enough to form the fundamental basis for the general solution.
Yes, in special non-solution cases the Wronskian can vanish for independent functions. For solutions of a common linear ODE, however, a zero Wronskian reliably indicates dependence.
From Laplace transforms to eigenvectors, SEOWebChecker's maths and physics tools help you solve problems faster.