Homogeneous 1st order PDE?
This PDE has auxiliary system dx/1 = dy/(2xy^2) Solving this via separation of variables: 2x dx = dy/y^2 ==> x^2 = -1/y + C ===> x^2 + 1/y = C. So, a general solution is u(x, y) = f(x^2 + 1/y) for some function f. Double check: By the Chain Rule, u_x = 2x f '(x^2 + 1/y) and u_y = (-1/y^2) f '(x^2 + 1/y) So, u_x + 2xy^2 u_y = 0, as required. ------- I hope this helps!
What is the complementary function of a linear homogenous partial differential equation when the roots are complex?
I am not confident about it because it has been a long time since I studied Maths :) But the answer is: y=c1*e^ax cosbt + c2*e^ax sinbt if roots are a+bi & a-bi
How do I know if this equation is a homogeneous differential equation?
If you have y’ = f(x, y), then this is homogenous if f(tx, ty) = f(x, y)—that is, if you put tx’s and ty’s where x and y usually go, and the result is the initial function, then this differential equation is homogenous. Note: this only applies to first-order differential equations. Finally, in the context of higher-order differential equations, a homogenous differential equation is simply one that equals 0, as in virtually all areas of mathematics.
What are homogeneous differences and non-homogeneous difference equations?
A2A, thanks.“Non homogeneous” here means “having a nonzero additive constant (e.g., see the term [math]b[/math] in Linear difference equation - Wikipedia).
Find the general integral of (x-y)p+(y-x-z)q=z which passes through the circle x^2+y^2=1, z=1?
As I can understand, you want the general solution of this differential equation and the particular solution through the aforementioned circle. The corresponding symmetric system for the equation is dx/(x - y) = dy/(y - x - z) = dz/z and you need 2 independent first integrals. Adding all numerators and denominators we get (dx + dy + dz)/(x - y + y - x - z + z) = d(x + y + z)/0 what means d(x + y + z) = 0, so the 1st independent integral is f₁ = x + y + z. Next (dx - dy + dz)/(x - y - y + x + z + z) = dz/z yields d(x - y + z)/(x - y + z) = 2 dz/z or d(ln(|x - y + z|/z²)) = 0 and we get the 2nd independent integral f₂ = (x - y + z)/z² According the theory of 1st order linear non-homogeneous partial differential equations the general solution in implicit form is Φ((x - y + z)/z², x + y + z) = 0, here Φ is a differentiable function of 2 variables, or (x - y + z)/z² = φ(x + y + z), where φ is an arbitrary differentiable function of 1 variable. Now we must determine φ to satisfy the initial condition. For z = 1 we get x - y + 1 = φ(x + y + 1). If x + y + 1 = t and y = ±√(1 - x²), then φ(t) = 1 ± √(1 + 2t - t²), finally (x - y + z)/z² = 1 ± √(1 + 2(x+y+z) - (x+y+z)²) or (x - y +z - z²)² = z⁴(1 + 2(x + y + z) - (x + y + z)²) in implicit form.
What makes Fully Nonlinear second order pde?
There is no mathematical word like fully linear.there is nonlinear or semiliner pde.
What is the degree and order of differential equations?
The order of a differential equation is the order of the highest order derivative involved in the differential equation. The degree of a differential equation is the exponent of the highest order derivative involved in the differential equation when the differential equation satisfies the following conditions –All of the derivatives in the equation are free from fractional powers, positive as well as negative if any.There is no involvement of the derivatives in any fraction.There shouldn’t be involvement of highest order derivative as a transcendental function, trigonometric or exponential, etc. The coefficient of any term containing the highest order derivative should just be a function of x, y, or some lower order derivative.For more information please watch the below video :
How do you solve the PDE u_x + u_y =1?
Note that this PDE is linear. Homogeneous solution u_h: du/dx + du/dy = 0 We get the characteristic curves via the ODE's dx/1 = dy/1 Solving this via integration yields x - y = C. So, u_h(x,y) = f(x - y), where f is an arbitrary function. Particular solution u_p: It is easy to verify that u_p(x,y) = x is one such solution. General solution: u(x,y) = f(x - y) + x. I hope that helps!
What are the real life applications of first order differential equations?
Applications of First-order Differential Equations to Real World Systems4.1 Cooling/Warming Lawthe mathematical formulation of Newton’s empirical law of cooling of an object in given by the linear first-order differential equation4.2 Population Growth and DecaydN(t)/dt =kN(t)where N(t) denotes population at time t and k is a constant of proportionality, serves as a model for population4.3 Radio-Active Decay and Carbon Datinga radioactive substance decomposes at a rate proportional to its mass. This rate is called the decay rate. If m(t) represents the mass of a substance at any time, then the decay rate dm/dt is proportionalto m(t). Let us recall that the half-life of a substance is the amount of time for it to decay to one-half of its initial mass.4.4 Mixture of Two Salt Solutions4.5 Series CircuitsThere are many applications of differential equations indeed!!