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Magazine Chapter 1: Problem 12
$$ x y^{\prime}+3 y=2 x^{5}, y(2)=1 $$
Short Answer
Expert verified
The particular solution is \( y = \frac{2}{9} x^6 - \frac{952}{9x^3} \).
Step by step solution
01
Identify the Type of Differential Equation
The given equation is in the form \( x y' + 3y = 2x^5 \). This is a first-order linear differential equation, where \( P(x) = \frac{3}{x} \) and \( Q(x) = \frac{2x^4}{x} = 2x^4 \).
02
Find the Integrating Factor
To solve the linear differential equation, we need to find the integrating factor \( \mu(x) \). It is given by the formula: \( \mu(x) = e^{\int P(x) \: dx} = e^{\int \frac{3}{x} \: dx} \). Calculating the integral, \( \int \frac{3}{x} \: dx = 3 \ln |x| \). The integrating factor is therefore \( \mu(x) = e^{3 \ln |x|} = x^3 \).
03
Apply Integrating Factor
Multiply every term of the differential equation by the integrating factor \( x^3 \): \( x^3(x y') + 3x^2 y = 2x^8 \). This simplifies to \( x^4 y' + 3x^3 y = 2x^8 \). With the integrating factor, the equation can be expressed as \( \frac{d}{dx}(x^3 y) = 2x^8 \).
04
Integrate Both Sides
Integrate both sides with respect to \( x \): \( \int \frac{d}{dx}(x^3 y) \, dx = \int 2x^8 \, dx \). This yields \( x^3 y = \frac{2}{9}x^9 + C \), where \( C \) is the constant of integration.
05
Solve for \( y \) (General Solution)
Divide both sides by \( x^3 \) to solve for \( y \): \( y = \frac{2}{9} x^6 + \frac{C}{x^3} \). This is the general solution of the differential equation.
06
Apply Initial Condition
Use the initial condition \( y(2) = 1 \) to find \( C \). Substitute \( x = 2 \) and \( y = 1 \) into the general solution: \( 1 = \frac{2}{9} (2)^6 + \frac{C}{(2)^3} \), which simplifies to \( 1 = \frac{128}{9} + \frac{C}{8} \).
07
Solve for the Constant \( C \)
Simplify and solve for the constant \( C \): \[ 1 - \frac{128}{9} = \frac{C}{8} \]. This implies \( -\frac{119}{9} = \frac{C}{8} \), so \( C = -\frac{119}{9} \times 8 = -\frac{952}{9} \).
08
Write the Particular Solution
Substitute the value of \( C \) back into the general solution to get the particular solution: \( y = \frac{2}{9} x^6 - \frac{952}{9x^3} \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Integrating Factor
When you encounter a first-order linear differential equation like the given one, the integrating factor is a crucial component in finding its solution. This technique is commonly used to transform the differential equation into a form that is easier to solve.
- The integrating factor, often denoted as \( \mu(x) \), is a function chosen to simplify the equation. It is always derived from the function \( P(x) \) in the standard form of our differential equation, \( y' + P(x)y = Q(x) \).
- In this scenario, we calculated it using the exponential of the integral of \( P(x) \), specifically: \( \mu(x) = e^{ \int P(x) \, dx} \). Therefore, the integrating factor becomes \( \mu(x) = x^3 \).
General Solution
Once the integrating factor is applied, the next task is to derive the equation's general solution. This involves rewriting the modified equation from the integration step.
- The differential equation becomes an easily integrable form after applying the integrating factor, presenting as the derivative of a product. For example, it appears as \( \frac{d}{dx}(x^3 y) = 2x^8 \).
- Integrating both sides with respect to \( x \), we find \( x^3 y = \frac{2}{9}x^9 + C \), where \( C \) is the integration constant. This allows us to solve for \( y \) directly, giving us the general solution: \( y = \frac{2}{9}x^6 + \frac{C}{x^3} \).
Initial Condition Application
With the general solution in hand, incorporating the initial condition is the final step in finding the specific or particular solution that satisfies the original problem.
- An initial condition such as \( y(2) = 1 \) provides a particular value of \( y \) at a specific \( x \). This condition is used to determine the exact value of the constant \( C \).
- We substitute \( x = 2 \) and \( y = 1 \) into the general solution to solve for \( C \): \( 1 = \frac{2}{9}(2)^6 + \frac{C}{2^3} \). Solving this gives \( C = -\frac{952}{9} \).
- By substituting \( C \) back into the general solution, we derive the particular solution that fits the initial condition exactly: \( y = \frac{2}{9}x^6 - \frac{952}{9x^3} \).
