Integrating an ODE

Question

ODE: x(dt/dx)=C, where 𝐶 is constant, in the domain 𝑥₁ ≤ 𝑥 ≤ 𝑥₂, subject to boundary conditions: 𝑇(𝑥₁ ) = 𝑇₁ and 𝑇(𝑥₂ ) = 𝑇₂. Find the dimensionless temperature distribution 𝑇(𝑥)−𝑇1 𝑇2−𝑇1 .

Looking for temperature distribution by finding T(x) from the ODE given.

There is a figure that goes along with this, but for some reason I can't add it, so if you think it would be helpful, please message me/ let me know.

Thank you

Answer 1

Substitute θ = (T-T1)/(T2 - T1) into diff eqn. note that the BCs are θ(x1)=0 and θ(x2) = 1

(T2-T1)(x)(dθ/dx) = C or xdθ/dx = C'

Separating and integrating from θ = 0 to θ and x from x1 to x (Yes, there should be dummy variables in the integrand)

Int(0 to θ) (dθ) = C' Int(x1 to x) (1/x)dx

θ = C' ln(x/x1)

Plug in θ = 1 for x = x2 and solve for C'

Write solution.