Several types of ordinary differential equations $($ODE$)$ will be used so frequently in ENME$321$ that reviewing the solution method is necessary. Again, the dependent variable, T represents temperature that varies in a $1$D space. Solve the following differential equations to obtain symbolic solution $T(x).$

$A,B,C,L,$ and $M$ are known constants. You are required to use indefinite integration when solving the ODEs, and use the boundary conditions to determine the unknown constant values. $c,d,$edots represents unknown constants to be determined in the solution process.

$d^2\frac{T}{d}{x}^2=0,$ at $x=0,T(0)=A,$ at $x=L,T(L)=B$

$d^2\frac{T}{d}{x}^2=0,$ at $x=0,T(0)=A,$ at $x=L,-d\frac{T}{d}x=T$

$d^2\frac{T}{d}{x}^2+C=0,$ at $x=0,T(0)=A,$ and at $x=L,T(L)=B$

$d\frac{T}{d}t=-AT,$ at $t=0,T=B,$

$d^2\frac{T}{d}{x}^2-m^{2}T=0,$ at $x=0,T(0)=A,$ at $x=L,T(L)=B$

This is a typical ODE with the general solution in the form of

$T=c{e}^{mx}+d{e}^{-mx}$

Applying BCs:

$A=c{e}^{m0}+d{e}^{-m0}A=c+d$

$B=c{e}^{mL}+d{e}^{-mL}B=c{e}^{mL}+d{e}^{-mL}$

Solving these equations for $c$ and $d$ yields:

$T(x)=\frac{\frac{B}{A}sinh(mx)+sinhm(L-x)}{sinh(mL)}$