Questions tagged [characteristics]

The method of characteristics is a way of solving certain partial differential equations by reducing them to ordinary differential equations. It is most often used for 1st order equations. Use with the (pde) tag.

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General solution of a first order PDE with zeroth order term

So I have got the following equation: $$x\frac{\partial u}{\partial x} - 2 \frac{\partial u}{\partial y} = 2u$$ I have tried to solve the following way. I was taught that LHS can be thought of as the ...
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Finding the characteristic curves of a PDE

The Question: In the region $y>0$, reduce the PDE $$y \frac{\partial ^2u}{\partial x^2} = \frac{\partial ^2u}{\partial y^2}$$ to canonical form, and sketch the characteristic curves. My Attempt:...
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Existence and uniqueness of PDE IVP : $z z_x + y z_y = x$, $C : x=t, y=t \; ; \; t >0$ and $z=t$ over $C$.

Exercise : Consider the equation $$z z_x + y z_y = x$$ and the initial curve $$C : x=t, y=t \; ; \; t >0$$ Decide whether there is a unique solution, no solution or infinitely many ...
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Unique general solution for linear PDE with method of characteristic

Consider the linear PDE $y u_x + x u_y = 0$. Applying the method of characteristics, we find that a general solution is given by $u(x,y) = f(x^2 - y^2)$ for some function $f$. My question is: can ...
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Principal Minor and Non-zero Eigen values in Diagonal Matrix

While reading the definition of pseudo determinant in here, I've found the following : $$pdet(L) : = \sum_{I\in[n] , |I| = r}det(L_{I,\ I}) = \prod_{i=1}^r\lambda_i$$ where $L_{I,\ I}$ denotes the ...
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I am currently trying to solve a problem I have already solved, but am trying to solve it the way our professor solved it. The PDE is given by: $$yu_x-2xyu_y=2xu, \>\>\>\>\>\>\>... 1answer 304 views Method of Characteristics(Advection equation with initial and boundary condition) Solve, using the Method of Characteristics, the equation \frac{\partial \rho}{\partial t} + \frac{\partial \rho}{\partial x}=-\mu\rho for x,t>0 with the conditions \rho(x,0)=f(x) and \rho(0,... 1answer 128 views Using method of characteristics to find general solution of PDE x^3 u_x + y u_x = 4 + 2x^2u Use the method of characteristics to find the general solution u(x; y) of the partial differential equation$$ x^3 \frac{\partial u}{\partial x} + y \frac{\partial u}{\partial x} = 4 + 2 x^2 u 
The question is solve the following partial differential equation using the method of characteristic $\frac{\partial{u}}{\partial{x}}-\frac{\partial{u}}{\partial{t}}=2t$, with the condition \$u=u(x,t)...