Question about how to solve an inseperable differential equation

Question

In my differential equations class we are discussing how to find the amount of salt at a specific time given the rate of change equation $$\frac{dS}{dT} = 2-\frac{S}{(15+T)}$$ where $T$ is time and $S$ is the amount of salt. I had the idea to attempt to rewrite the equation so that we can take the integral of each part of the equation. I encountered a problem where $T$ didn't always align with the $dT$ and similar for $S$. I decided to partially integrate, similar to taking a partial derivative and got an answer that seemed right. I want to know if it is viable to use this method to solve an inseparable equation or if I would need to use other methods in order to arrive at the correct answer.

Answer 1

Just separation of variables will not suffice, but you will use it at one point.

When i encounter a new differential equation, the first thing i think about is its type (like linear, nonlinear, autonomous, homogeneous, etc). So think about what type your equation is and what methods apply.

Answer 2

You cannot directly apply separation of variables.

This is a first-order linear ordinary differential equation: $$\frac{dS}{dT}+f(T)S=g(T)$$ Put your equation in the form above: $$\frac{dS}{dT}+\frac{1}{15+T}S=2 \tag{1}$$ You can solve this using an Integrating factor or by using Variation of parameters.

For this example, I will use the integrating factor method.

The integrating factor will be given by: $\mu(T)=e^{\int \frac{1}{T+15}~dT}=T+15$. No need to consider the arbitrary constant of integration.

Therefore, we multiply this by $\mu(T)$ (Our integrating factor) on both sides of $(1)$: $$(T+15)\frac{dS}{dT}+S=2(T+15)\tag{2}$$ Now, substitute $1=\frac{d}{dT}(T+15)$: $$(T+15)\frac{dS}{dT}+\frac{d(T+15)}{dT}S=2(T+15)\tag{3}$$ And use the reverse chain rule on the LHS of equation $(3)$: $uv'+v'u=(uv)'$. Then integrate both sides with respect to $T$.

Feel free to ask questions if you cannot continue from here.