Solve the differential equiation y'-2y=1, then calculate y(0) assuming the constant of integration is 2. Round the solution to the first decimal.

To solve the differential equation \( y' - 2y = 1 \), we'll follow these steps:

### Step 1: Solve the homogeneous equation.
The homogeneous equation associated with \( y' - 2y = 1 \) is:
\[
y' - 2y = 0
\]
This can be rewritten as:
\[
\frac{dy}{dx} = 2y
\]
We can separate variables and integrate:
\[
\frac{dy}{y} = 2 \, dx
\]
Integrating both sides:
\[
\ln |y| = 2x + C_1
\]
Exponentiating both sides to solve for \( y \):
\[
y = C_2 e^{2x}
\]
where \( C_2 = e^{C_1} \) is the constant of integration.

### Step 2: Solve the non-homogeneous equation.
We use the method of undetermined coefficients. Since the non-homogeneous term is \( 1 \), we assume a particular solution of the form \( y_p = A \), where \( A \) is a constant.

Substituting \( y_p = A \) into the differential equation:
\[
0 - 2A = 1
\]
\[
-2A = 1 \implies A = -\frac{1}{2}
\]
So, the particular solution is \( y_p = -\frac{1}{2} \).

### Step 3: Combine the solutions.
The general solution to the differential equation is:
\[
y(x) = y_h + y_p = C_2 e^{2x} - \frac{1}{2}
\]

### Step 4: Apply the initial condition.
We are given that the constant of integration \( C_2 \) is 2. So, we substitute \( C_2 = 2 \) into the general solution:
\[
y(x) = 2e^{2x} - \frac{1}{2}
\]

### Step 5: Calculate \( y(0) \).
Substituting \( x = 0 \) into the equation:
\[
y(0) = 2e^{0} - \frac{1}{2} = 2(1) - \frac{1}{2} = 2 - 0.5 = 1.5
\]

So, \( y(0) = 1.5 \).

**Rounded to the first decimal place, \( y(0) = 1.5 \)**.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How do you find the general solution of a first-order linear differential equation?
2. What are the steps involved in solving non-homogeneous differential equations using the method of undetermined coefficients?
3. How does the particular solution differ from the homogeneous solution in a differential equation?
4. What role does the constant of integration play in the solution of differential equations?
5. Can you solve the differential equation \( y' + 3y = 2 \) using a similar method?

**Tip:** In differential equations, verifying the solution by substituting it back into the original equation is a good practice to ensure accuracy.

Learn how to solve the differential equation y'-2y=1 and calculate y(0) with a detailed step-by-step solution. This problem covers differential equations, first-order linear equations, homogeneous equations, and the method of undetermined coefficients. Suitable for undergraduate-level mathematics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation