\[ y^{\prime\prime}(t) + p(t) y^{\prime}(t) + q(t) y(t) = g(t) \]
The functions $p(t), q(t)$ and $g(t)$ are called coefficients. The coefficient $g(t)$ is called the forcing term. If $g(t)=0$, then the equation is said to be homogeneous:
\[ y^{\prime\prime}(t) + p(t) y^{\prime}(t) + q(t) y(t) = 0 \]
The Wronskian of functions $f(t)$ and $g(t)$ is the following function:
$ W(f,g)(t) = f(t)g^{\prime}(t) - f^{\prime}(t) g(t) $
If the Wronskian is non-zero at some point in an interval, then the functions $f(t)$ and $g(t)$ are linearly independent on the interval.
Two functions $y_1(t)$ and $y_2(t)$ form a fundamental set of solutions of a homogeneous equation
\[ y^{\prime\prime}(t) + p(t) y^{\prime}(t) + q(t) y(t) = 0 \]
$y_1(t)$ and $y_2(t)$ are solutions of the homogeneous equation $y^{\prime\prime} + py^{\prime} + qy = 0$.
The Wronskian $W(y_1,y_2)(t) = y_1(t)y_2^{\prime}(t) - y_1^{\prime}(t) y_2(t)$ is nonzero.
When we are solving a homogeneous equation
\[ y^{\prime\prime}(t) + p(t) y^{\prime}(t) + q(t) y(t) = 0, \]
we first find a fundamental set of solutions $y_1(t)$ and $y_2(t)$. Then all the solutions are given as linear combinations of $y_1(t)$ and $y_2(t)$, that is as,
$ C_1 y_1(t) + C_2 y_2(t). $
This formula is the general solution of the homogeneous equation. Here $C_1$ and $C_2$ are arbitrary constants.
\[ y^{\prime\prime}(t) + p(t) y^{\prime}(t) + q(t) y(t) = 0, \ \ \ \ y(0) = y_0, \ \ \ y^{\prime}(0) = v_0, \]
we first find the general solution, then we calculate $C_1$ and $C_2$ from the equations
$C_1 y_1(0) + C_2 y_2(0) = y_0$ and $C_1 y_1^{\prime}(0) + C_2 y_2^{\prime}(0) = v_0.$
This gives us the particular solution which satisfies the initial conditions.
Remember, $V$ is the volume of the tank, $a$ gal/min of brine enters the tank and the brine leaves the tank with the same rate, $c$ is the concentration of salt in the incoming brine in lb/gal. Denote by $S(t)$ the amount in lb of salt in the tank at time $t$. The differential equation governing this process is
\[ S'(t) = c\, a - \frac{a}{V} \, S(t). \]
The common sense tells us that if there are $S(0) = c\,V$ pounds of salt in the tank at time $t = 0,$ then there will be no change in the amount of salt in the tank. That is $S(t) = c\,V$ is an equilibrium solution. Again, common sense tells us that if $S(0) > c\,V$, then $S(t)$ will decrease and in the limit it will get closer and closer to $c\,V$. Similarly, if $S(0) < c\,V$, then $S(t)$ will increase and in the limit it will get closer and closer to $c\,V.$ It might be informative to rewrite the differential equation as
\[ S' = \frac{a}{V} \bigl( c\, V - S \bigr). \]
From this equation we can see that $S'$ is positive, that is the function $S$ is increasing, whenever $S < c\,V$ (in this case $c\,V - S > 0$). Similarly, $S'$ is negative, that is the function $S$ is decreasing, if $S > c\,V$ (in this case $c\,V - S < 0$).
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