# Questions tagged [wronskian]

This tag is for various questions relating to "Wronskian". In mathematics, it is a determinant used in the study of differential equations, where it can sometimes show linear independence in a set of solutions.

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### Wronskian of second order differential equation $ty'' − (t + 1)y' − y = 0$.

Find a Wronskian of two solutions of $$ty'' − (t + 1)y' − y = 0, ~~~t > 0$$ provided $W[y_1, y_2](1) = 1$. Answer: $W [y_1, y_2] (t) = te^{t−1}$ I am unsure of how they got the answer, am I ...
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### When using the Frobenius method, and r1-r2 is neither zero nor a positive integer, can you use the Wronskian to find the second solution?

When using the Frobenius method, and r1-r2 is neither zero nor a positive integer, can you use the Wronskian to find the second solution? Basically, do I have to repeat the substitution with the ...
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### Substituting $t_0$ in Wronskian to solve 2nd order non homogeneous ode?

Hey everyone I need some help understanding how to use Lagrange method and specifically the Wronskian used in it to solve 2nd order non homogeneous ode. Let : $$y''(t)+a_1y'(t)+a_2y(t)=f(t)$$ I ...
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### any two solution of the equation $y''+p(x)y'+q(x)y=0$, $p(x)$ and $q(x)$ are continuous on $(a,b)$ and $x\in (a,b)$ are linearly dependent

Any two solution of the equation $y''+p(x)y'+q(x)y=0$, $p(x)$ and $q(x)$ are continuous on $(a,b)$ and $x\in (a,b)$ are linearly dependent if (a) they have common zero in $(a,b)$ (b) they have a ...
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### Wronskian and Linear Dependence

Let $y_1$, $y_2$ be two solutions of a homogeneous linear second order differential equation $y^{''}$ + $p(t)y^{'}$ + $q(t)y$ = $0$ over the interval $\alpha$ < $t$ < $\beta$. Prove that ...
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### Construct a Second Order ODE given the fundamental solutions

I need to construct a second order linear differential equation for which $\{ \sin (x), x \sin (x) \}$ is the set of fundamental solutions. I am completely lost on this problem and have been trying ...
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### How to prove a Wronskian identity?

The following Wronskian identity can be proved by expanding both sides and checking that two sides are the same. But how to prove it more elegantly? Let $u_1(x), u_2(x), u_3(x), u_4(x)$ be four ...
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### Linear independence and the Wronskian

I want to show linear independence in the wronskian implies linear independence between the functions $f_1(x)$, $f_2(x)$, $f_3(x)$. Let $f_1(x)$, $f_2(x)$, $f_3(x)$ be real-valued functions with first ...
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### Linear dependance and Wronskian determinant

I am asked to show that the fuctions $e^x, \cos(x) \text{ and } x^2$ are linearly independent. I wanted to use the Wronskian determinant in order to prove the above property. We have: W= \begin{...