Classify the following partial differential equations:(i) 𝜕2𝑢𝜕𝑥2 + 3 𝜕2𝑢𝜕𝑦2 = 𝜕𝑢
Question
Classify the following partial differential equations:
(i)
Solution
The given partial differential equation is:
𝜕²𝑢/𝜕𝑥² + 3 𝜕²𝑢/𝜕𝑦² = 𝜕𝑢
To classify this PDE, we need to look at the highest order derivatives. In this case, we have second order derivatives with respect to x and y.
The general form of a second order PDE is:
A(𝜕²𝑢/𝜕𝑥²) + B(𝜕²𝑢/𝜕𝑦²) + C(𝜕𝑢/𝜕𝑥) + D(𝜕𝑢/𝜕𝑦) + E𝑢 = F
Where A, B, C, D, E are coefficients which can be functions of x and y.
The classification of the PDE depends on the discriminant B² - 4AC.
In our case, A = 1, B = 3, C = 0, D = 0, E = 0. So, the discriminant is B² - 4AC = 3² - 410 = 9.
Since the discriminant is positive, the given PDE is an elliptic type.
Similar Questions
Classify the following partial differential equations:(i) 𝜕2𝑢𝜕𝑥2 + 3 𝜕2𝑢𝜕𝑦2 = 𝜕𝑢
Identify the type of the differential equation (as linear, homogeneous,exact, separable or Bernoulli) and find its general solutiondxdt = 2x + etx32
If 𝑧=𝑠𝑖𝑛(3𝑥+2𝑦), find its derivative with respect to xQuestion 7Answera.𝑐𝑜𝑠(3+2𝑦)b.3𝑐𝑜𝑠(3𝑥+2𝑦)c.𝑠𝑖𝑛(3+2𝑦)d.3𝑠𝑖𝑛(3+2𝑦)
Use Lagrange method to write the general solution of the following partial differentialequations.
A system comprising three standard ordinary differential equations (ODEs) has been successfully resolved.
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.