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Let us consider a differential equation of the form :

$$\frac{\partial^2 u}{\partial x^2} + a\frac{\partial u}{\partial x} = 0$$ The solution is of the form $u = c_0 + c_1\exp(ax)$. Wtih boundary conditions $u(\infty) = 0.5$ and $u(0) = -bV$. Here, b and V are constants. How do I now calculate the exact solution of this differential equation ?

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This is incorrect. The auxiliary equation formed would be $$m^2+am=0$$ $$m(m+a)=0$$ $$m=0,\, -a$$ $$\therefore u=c_0+c_1 e^{-ax}$$ Assuming that $a\gt 0$ and $a\in \mathbb{R}$, $$u(\infty)=c_0=0.5$$ But one cannot determine the exact solution without another boundary condition. So the final solution is then $$u=0.5+c e^{-ax}$$

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  • $\begingroup$ @Terence, don't add things to the answer that wasn't the author's intent. If you think there are steps missing, write a comment, or better, post your own answer. $\endgroup$ – Dylan Mar 5 at 9:10
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As pointed out in another answer, the generic solution to

$\frac{d^2 u}{d x^2} + a\frac{d u}{d x} = 0$

for $a \ne 0$ is actually

$u = c_0 + c_1\exp(-ax)$

As long as $a>0$ then we have

$u(0) = c_0 \\ u(\infty) = c_0+c_1 \\ \Rightarrow c_1 = u(\infty) - u(0)$

If $a<0$ then $u(\infty)$ is undefined as $u$ is unbounded as $x \rightarrow \infty$.

If $a=0$ then the generic solution is $u=c_0+c_1x$ and again $u(\infty)$ is undefined.

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