Thermal Expansion and Differential Equations

[temaire]

Homework Statement

Homework Equations

$$\alpha(T)dT = \frac{\partial L}{L}$$ <--- Differential Equation given on formula sheet

The Attempt at a Solution

$$\alpha(T)dT = \frac{\partial L}{L}$$

$$\int_{l_0}^{l_1} \frac{\partial L}{L} = \int_{T_0}^{T_1} \alpha(T)dT$$

$$ln(\frac{l_1}{l_0}) = \int_{T_0}^{T_1} [ \alpha_{0} + \alpha_{1} T]dT$$

$$ln(\frac{l_1}{l_0}) = \alpha_{0} (T_1 - T_0) + \frac{\alpha_{1} (T_1 - T_0)^{2}}{2}$$

$$\frac{l_1}{l_0} = e^{\alpha_{0} (T_1 - T_0) + \frac{\alpha_{1} (T_1 - T_0)^{2}}{2}}$$

I know my answer is not correct, since the question asked to have the equation in terms of $l$. I'm also confused as to whether or not I was even supposed to use the equation that was given in the preamble, $\alpha(K) = \alpha_{0} + \alpha_{1} T$. I would appreciate any help.

 

[Valkarie]

The correct solution should be:\frac{l_1}{l_0} = e^{\alpha_{0}(T_1 - T_0)}e^{\frac{\alpha_{1}}{2}(T_1^2 - T_0^2)}