Integrating something that already has a dx in it

Question

What would integrating both sides of x*(dT/dx)=C in terms of x look like?

I keep getting that T(x)=C, but i don't think that's right.

boundary conditions: 𝑇(𝑥₁ ) = 𝑇₁ and 𝑇(𝑥₂ ) = 𝑇₂

Thank you

Answer 1

More straightforward than directly integrating both sides is to divide by x first, then integrate both sides

This leads to T(x) = C ln(x) + const.

T₁ = C ln(x₁) + const

T₂ = C ln(x₂) + const

So T₂ - T₁ =C ln(x₂/x₁) ; so C = (T₂ - T₁) / ln(x₂/ x₁)

and const = T₁ - ( T₂ - T₁) [ ln(x₁) / ln(x₂/ x₁) ]