# Wronskian

### Wronskian

#### Lessons

A linear homogeneous second order differential equation is of the form:
$a(x) y''+b(x) y'+c(x)y=0$

Let’s assume that we’ve found two solutions to the above differential equation, $y_1 (x)=f(x)$ and $y_2 (x)=g(x)$.

And the general solution will be of the form:
$y(x)=c_1 f(x)+c_2 g(x)$

We can us the Wronskian to see whether $f(x)$ and $g(x)$ are linearly independent

The Wronskian is defined as:

If we have two solution, $f(x)$ and $g(x)$, and $W(f,g)\neq0$, then we say that $f(x)$ and $g(x)$ form a fundamental set of solutions, and the general solution will indeed be of the form:

$y(x)=c_1 f(x)+c_2 g(x)$
• 1.
How can we be sure that our general set of solutions is indeed the general set of solutions? The Wronskian!

• 2.
Verifying Some of Our General Solutions
In the section “Characteristic Equation with Repeated Roots” we had the following differential equation:
$y''-6y'+9y=0$

It was found that the two solutions were:
$y_1 (x)=e^{3x}$
$y_2 (x)=xe^{3x}$
a)
But one might have made the assumption that both solutions were of the form: $y_1 (x)=y_2 (x)=e^{3x}$. Demonstrate that these solutions are not linearly independent

b)
Show that the two actual solutions actually form a fundamental set of solutions.

• 3.
In the section “Characteristic Equation with Real Distinct Roots” we had the following differential equation:
$6y''+8y'-8y=0$
We found two solutions:

$y_1 (x)=e^{\frac{2}{3} x}$
$y_2 (x)=e^{-2x}$

Verify that these two solutions are indeed linearly independent

• 4.
A certain differential equation was found to have two solutions:
$y_1(x)=3 \cos (2x)$
$y_2 (x)=3-6\sin^2 (x)$
Are these two solutions independent? Will they form a fundamental set of solutions?