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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What is a Wronskian and why is it useful here?
A particle in the region $z$ is described by $\psi(z, p)=A(p) f(z, p)$. Where $A(p)$ is independent of $z$ and $f(z,p)$ is the solution to:
\begin{equation}
\left[\frac{\partial^2}{\partial z^2}+\frac{...
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Are $\tau$ and $t$ given as boundary/initial conditions? If not, how to find both?
Recall from the Fundamental Sets and Matrices of a Linear Homogeneous nth Order ODE page that if we have a linear homogeneous $n^{\text {th }}$ order ODE $y^{(n)}+a_{n-1}(t) y^{(n-1)}+\ldots+a_1(t) y^{...
user1317687
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Where does this expression for Green's function come from?
In the context of $2$nd order ODEs, I found in some solution sheet that they computed the Green's function using the following expression
$$G(x;x')=\dfrac{y_2(x)y_1(x')-y_1(x)y_2(x')}{\overline{\...
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Is this a valid formula for the Wronskian?
I was messing around with the Wronskian of two functions $y_1(x)$ and $y_2(x)$, which is defined by:
$$
W(y_1,y_2) = \begin{vmatrix}
y_1 & y_2 \\
y_1' & y_2'
\end{vmatrix} = y_1y_2'-y_2y_1^{\...
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Differential equation by the parameter variation method
I need help to solve this equation by the method of parameters variation (Wronskian):
$$4\,y''-3\,y'=x\,e^{\frac{3}{4}\,x}$$
I know the solution of the differential equation is:
$$y=\dfrac{{x}^{2}\,{e}...
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Introduce $y(x)=u(x)z(x)$ into the equation $y''-2xy'-2y=0$ so that there won't be a term with $z'$ in the new equation.
Introduce $y(x)=u(x)z(x)$ into the equation $y''-2xy'-2y=0$ so that there won't be a term with $z'$ in the new equation. Find all solutions of such equations and also explore the possibility when $z'=...
user1315539
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Why only test Wronskian at single time t rather than over the entire interval the solution exists
When determining linear independence of solutions to a linear ODE, why do we only need to test the Wronskian at a single time t, rather than over the entire interval in which the solution exists?
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How can I solve a Bessel equation with Reduction of order?
If $y_{1}(x) = \frac{\sin(x)}{\sqrt(x)}$ is one solution of the differential equation $$x^2y'' +xy' + (x^2-\frac{1}{4})y = 0$$
find the second solution $y_{2}(x)$.
My effort using Wronskian
The ...
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Proving solutions of $y''+p(x)y'+q(x)y=0$ to be linearly independent
When studying Elementary Differential Equations by William, I found trouble understanding
Theorem 5.1.5
It says the two solutions are linearly independent iff their Wronskian is never zero, but I ...
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Wronskian & Linear independence of functions
We usually study wronskian concerned with solutions of ODE, but using this idea to check linear dependence of any two differentiable functions seems Okay because it is the determinant of a Matrix ...
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Find general solution of Differential equation if you know three solutions. Is there exist general solution if Wronskian is zero? [closed]
Find general solution of Differential equation if you know three solutions. I tried to solve this problem, however I have a question about the Wronskian. Three particular solutions are 1, $x$, ${x^2}$....
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Derivative of a rational function on a curve with respect to a nonconstant function
Let $C$ be a smooth projective curve over an algebraically closed field $k$, and let $k(C)$ be its function field. For elements $f_1,...,f_n \in k(C)$ and a nonconstant $t \in k(C) \setminus k$, we ...
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Is this set of functions linearly dependent or independent? [duplicate]
The given functions are solutions to a differential equation
\begin{equation*} y_1(x)=\cos(2x),y_2(x)=1,\;y_3(x)=\cos(x) \end{equation*}
I need help determining if the set of functions are linearly ...
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Prove each non-trivial solution, $u(x)$ of a normal, second-order linear ordinary differential equation, has only simple zeroes.
Prove each non-trivial solution, $u(x)$ of a normal, second-order linear ordinary differential equation,
$$a_2(x)y'' + a_1(x)y' + a_0(x)y = 0$$
$\forall\ x \in I$
has only simple zeroes.
Where point $...
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How can we use the initial condition $y(0)=0$, the solution is not even defined at $0$?
The question states:
Consider the differential equation $x^2y''+xy'-y=0$. If $y_1$ and $y_2$ are two linearly independent solutions to the differential equation then choose the incorrect:
(1) $W(y_1,...
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