Differential Equations Answers

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

2(y+z)dx-(x+z)dy+(2y-x+z)dz=0