Differential Equations Answers

Determine the steady-state temperature u(x,y) in a rectangular plate A(0,0), B(0,6), C(10,6), D(10,0) if the temperature along AB, BC, and AD are maintained at zero while edge CD is at temperature 80. Note that the flat faces are insulated.

A thin, homogenous bar of length L has insulated ends and initial temperature f(x) = x. Determine the temperature distribution u(x,t) in the bar.

Find conditions under which a scalar conservation law can have a stationary jump discontinuity.

Given that z = ax + √(a² − 4y) + c is the complete integral of the PDE, p²-q² = 4 ,determine its general integral.

Find the differential equation of the space curve in which the two families of surfaces

u = x² − y² = c₁ and

v = y² − z² = c₂ intersect.

Find the partial differential equation arising from φ(z/x³, y/z) = 0

 where φ: R²→ R is an arbitrary function.

Also find the general solution of the PDE obtained.

Z=a(x+log y)-x^2/2-bx

Drive the following formulas;

d/dx [x^vY_v(x)]=x^vY_v-1(x)

Drive the following formulas;

[x^vY_v(x)]=x^vY_v-1(x)

find the differential equation 𝑦=2A𝑥^4+4B𝑥^3−6C𝑥^2+5D𝑥 +11E