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1. ### Heat transfer direction in fins

Homework Statement I'm unsure of what exactly is changing the heat transfer direction in the triangular fin. Homework Equations $$q_{x} = -kA(x)\frac{dT(x)}{dx} (1)$$ $$q_{x+dx} = -kA(x)\frac{dT(x)}{dx} - k\frac{d}{dx}[A(x)\frac{dT(x)}{dx}] (2)$$ $$dq_{conv} = h(x)dS(x)P[T(x) - T_{∞}] (3)$$...
2. ### Series solutions to ODE

Homework Statement Solve for xy'' + y' +αy + βxy = 0 α and β are constants The Attempt at a Solution What I initially had in mind was: xy'' + y' +αy + βxy = x²y'' + xy' +αxy + βx²y = 0 y = \sum_{n=0}^\infty a_n x^{n} xy = \sum_{n=0}^\infty a_n x^{n+1} = \sum_{n=1}^\infty a_{n-1} x^{n} = a_0x...
3. ### ODE by Laplace transform

Homework Statement Use Laplace transform to solve the following ODE Homework Equations xy'' + y' + 4xy = 0, y(0) = 3, y'(0) = 0 The Attempt at a Solution L(xy'') = -\frac{dL(y'')}{ds} L(4xy) = -\frac{4dL(y)}{ds} L(y'') = s²L(y) - sy(0) - y'(0) = s²L(y) -3s L(y') = sL(y) - sy(0) - y(0) =...
4. ### Fin/Extended surface differential equation for temperature

I'm trying to deduce the differential equation for temperature for a triangular fin: I know that for a rectangular fin, such as: I can do: Energy entering the left: q_x= -kA\frac{dT(x)}{dx} Energy leaving the right: q_{x+dx} = -kA\frac{dT(x)}{dx} - kA\frac{d² T(x)}{dx²}dx Energy lost by...