3.8 Q-21

Question Statement

Solve the differential equation:

$$\frac{dS}{dt} + 2 S t = 0$$

Also, find the particular solution when $S = 4e$ when $t = 0$.

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Background and Explanation

This is a first-order linear differential equation. To solve it, we will use the method of separating variables. This involves isolating the terms involving $S$ on one side of the equation and the terms involving $t$ on the other side. After integration, we will apply the initial condition to determine the particular solution.

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Solution

1. Rearrange the equation:

   Start with the given equation:

$$\frac{dS}{dt} + 2 St = 0$$

Move the term involving $S$ to the other side:

$$\frac{dS}{dt} = -2 S t$$

Now, we separate the variables $S$ and $t$ by dividing both sides by $S$:

$$\frac{1}{S} dS = -2 t , dt \tag{i}$$

2. Integrate both sides:

   Integrate both sides of equation (i). On the left side, we integrate with respect to $S$, and on the right side, we integrate with respect to $t$:

$$\int \frac{1}{S} , dS = -2 \int t , dt$$

The integrals give us:

$$\ln |S| = -t^2 + \ln C \tag{ii}$$

Where $C$ is the constant of integration.

3. Simplify the equation:

   Rearrange equation (ii) by isolating $\ln |S|$:

$$\ln |S| - \ln C = -t^2$$

Using properties of logarithms, combine the terms on the left-hand side:

$$\ln \left( \frac{S}{C} \right) = -t^2$$

Exponentiate both sides to eliminate the logarithm:

$$\frac{S}{C} = e^{-t^2}$$

Finally, multiply both sides by $C$ to get the expression for $S$:

$$S = C e^{-t^2}$$

4. Use the initial condition:

   The initial condition is $S = 4e$ when $t = 0$. Substitute these values into the equation $S = C e^{-t^2}$:

$$4e = C e^{0}$$

Simplifying:

$$4e = C \quad \Rightarrow \quad C = 4e$$

5. Write the particular solution:

   Now that we have the value of $C$, substitute it back into the equation for $S$:

$$S = 4e e^{-t^2}$$

Simplify the expression:

$$S = 4e^{1 - t^2}$$

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Key Formulas or Methods Used

• Separation of Variables: Rearranging the equation to separate the variables $S$ and $t$.
• Integration: Using basic integration rules to solve both sides of the equation.
• Initial Condition: Applying the given values of $S$ and $t$ to find the constant of integration.

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Summary of Steps

1. Start with the equation $\frac{dS}{dt} + 2 St = 0$.
2. Rearrange to $\frac{1}{S} dS = -2 t , dt$.
3. Integrate both sides: $\ln |S| = -t^2 + \ln C$.
4. Simplify to $S = C e^{-t^2}$.
5. Use the initial condition $S = 4e$ when $t = 0$ to find $C = 4e$.
6. The particular solution is $S = 4e^{1 - t^2}$.