线性微分方程水箱问题：初始条件x(0)=0的合理性疑问

Hey there! Let's break this down clearly since you've already mastered the modeling and integration steps—awesome work getting that far.

How to Solve for Constant C and Substitute Values #

Once you have the general solution to your linear differential equation (the one with the constant ( C ) in it), finding ( C ) just requires plugging in your initial condition:

• Identify your initial condition (like ( x(0) = 0 ) in your case). This tells you the value of ( x(t) ) at time ( t=0 ).
• Substitute ( t=0 ) into your general solution equation. All terms with ( t ) will simplify (since ( e^0 = 1 ), any term multiplied by ( t ) becomes 0, etc.).
• Rearrange the equation to solve for ( C ).
• Plug this calculated ( C ) back into the general solution to get the particular solution that fits your specific problem.

Example for Context #

Suppose your general solution looks like this (super common for tank problems):

x(t) = Vc_in(1 - e^{-kt}) + Ce^{-kt}

Where:

• ( V ) = total volume of the tank
• ( c_in ) = concentration of toxic substance in the inflow
• ( k = \frac{\text{flow rate}}{V} )

Using ( x(0) = 0 ):

• Substitute ( t=0 ): ( 0 = Vc_in(1 - e^{0}) + Ce^{0} )
• Simplify: ( 0 = Vc_in(0) + C(1) ) → ( C = 0 )
• Your particular solution becomes: ( x(t) = Vc_in(1 - e^{-kt}) )

Why ( x(0) = 0 ) Is a Reasonable Initial Condition #

The initial condition ( x(0) = 0 ) simply means:

At time ( t=0 ) (the starting point of our analysis), the tank contains no toxic substance at all.

This is a totally valid and common setup for tank problems, especially in textbook scenarios where:

• The tank starts filled with clean, pure water before any toxic solution begins flowing in.
• We choose ( t=0 ) to be the exact moment when the toxic inflow starts—so before that, there's no toxin present.

If your problem had a different starting state (e.g., the tank already had 50g of toxin at ( t=0 )), you'd use ( x(0) = 50 ) instead. The initial condition always matches the real-world state at your chosen time origin.

内容的提问来源于stack exchange，提问作者Stuy