3.8 Q-20

Question Statement

Solve the differential equation:

$$\frac{d y}{d t} = 2 x$$

Given that $x = 4$ when $t = 0$.

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Background and Explanation

This is a first-order linear differential equation. The equation involves $x$ and $t$, and we are tasked with finding $x$ as a function of $t$. To solve this, we will separate the variables and integrate both sides to obtain the general solution. We will then use the initial condition to find the constant of integration and determine the particular solution.

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Solution

1. Rearrange the equation:

   Start with the given equation:

$$\frac{d y}{d t} = 2 x$$

Notice that we have a relationship between $y$ and $x$ with respect to $t$. We need to separate variables, so we rewrite the equation as:

$$\frac{1}{x} , d x = 2 , d t \tag{i}$$

2. Integrate both sides:

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

$$\int \frac{1}{x} , d x = 2 \int d t$$

The integrals give us:

$$\ln x = 2 t + \ln C \tag{ii}$$

Where $C$ is the constant of integration.

3. Simplify the equation:

   We can simplify the equation by isolating $\ln x$:

$$\ln \frac{x}{C} = 2 t$$

Now, exponentiate both sides to remove the natural logarithm:

$$x = C e^{2 t}$$

4. Use the initial condition:

   We are given that $x = 4$ when $t = 0$. Substitute these values into the equation $x = C e^{2 t}$ to find $C$:

$$4 = C e^{2 \cdot 0}$$

Simplifying:

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

5. Write the particular solution:

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

$$x = 4 e^{2 t}$$

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Key Formulas or Methods Used

• Separation of Variables: Rewriting the equation to isolate terms involving $x$ on one side and terms involving $t$ on the other side.
• Integration: Using basic integration formulas to integrate both sides of the equation.
• Initial Condition: Using the given initial values to solve for the constant of integration $C$.

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Summary of Steps

1. Start with the equation $\frac{d y}{d t} = 2 x$.
2. Rearrange to separate variables: $\frac{1}{x} , d x = 2 , d t$.
3. Integrate both sides: $\ln x = 2 t + \ln C$.
4. Simplify to get $x = C e^{2 t}$.
5. Use the initial condition $x = 4$ when $t = 0$ to find $C = 4$.
6. The particular solution is $x = 4 e^{2 t}$.