Solve dp/dt= .5t-500 , a differential equation

In this video, we will learn how to solve the differential equation dp/dt = 0.5p - 500. First, we will multiply both sides by dt to get dp = 0.5p - 500 dt. Next, we will simplify the expression by factoring out 0.5 from the right-hand side to get dp = 0.5(p - 1000) dt. We can then rearrange this equation to get dp / (p - 1000) = 0.5 dt.

To solve for p, we will integrate both sides with respect to t. After integrating, we will use ln(p - 1000) to simplify the left-hand side of the equation. Then, we will exponentiate both sides to get p - 1000 = ± ae^(0.5t), where a is a constant. Finally, we will add 1000 to both sides to solve for p, giving us the equation p = ae^(0.5t) + 1000.

To find the value of a, we will use an initial condition. For example, if p = 2000 when t = 0, we can substitute these values into the equation and solve for a. After solving, we will have the specific equation p = e^(0.5t) + 1000.

🔎 Keywords differential equation, solve, initial condition, integration, exponentiation, constant

🔥 Tags #differential equation #solve #integration #exponentiation #mathematics #tutorial

📝 Common Questions

1️⃣ What is a differential equation?
A differential equation is a mathematical equation that relates a function to its derivatives.

2️⃣ How do you solve a differential equation?
To solve a differential equation, you need to find a function that satisfies the equation.

3️⃣ What is an initial condition?
An initial condition is a value that specifies the value of the function at a particular point in time.

4️⃣ What is integration?
Integration is a mathematical operation that calculates the area under a curve.

5️⃣ What is exponentiation?
Exponentiation is a mathematical operation that raises a number to a power.

6️⃣ What is a constant?
A constant is a fixed value that does not change.

7️⃣ How do you find the value of a constant in a differential equation?
To find the value of a constant in a differential equation, you need to use an initial condition.

8️⃣ What is ln?
ln is the natural logarithm function.

9️⃣ What is e?
e is a mathematical constant that is approximately equal to 2.71828.

🔟 What is the difference between e^x and 2^x?
e^x is the exponential function with base e, while 2^x is the exponential function with base 2.

1️⃣1️⃣ How do you solve for a differential equation with a constant?
To solve for a differential equation with a constant, you need to use an initial condition to find the value of the constant.

1️⃣2️⃣ What is a derivative?
A derivative is a mathematical concept that measures how much a function changes as its input changes.

1️⃣3️⃣ What is a function?
A function is a mathematical object that relates one input to one output.

1️⃣4️⃣ What is a curve?
A curve is a geometric object that can be defined by a mathematical equation.

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