Problem 48 - Calculus for Biology and Medicine

Question

Find the general solution of the differential equation. $$ \frac{d y}{d x}=\frac{2}{x^{3}}-x^{3}, x>0 $$

Step-by-Step Solution

Answer

The general solution is \( y(x) = -\frac{1}{x^2} - \frac{x^4}{4} + C \).

Step 1: Identify the Type of Differential Equation

The given differential equation is handwritten in the form \( \frac{dy}{dx} = \frac{2}{x^3} - x^3 \). This is a first-order linear differential equation in normal form, which is separable and can also be solved using integration.

Step 2: Separate Variables

Rearrange the terms to allow for separation of variables. With \( \frac{dy}{dx} = \frac{2}{x^3} - x^3 \), you can write this as \( dy = \left(\frac{2}{x^3} - x^3\right) dx \).

Step 3: Integrate Both Sides

Integrate both sides of the equation with respect to their respective variables. On the left side, integrate with respect to \( y \), and on the right side, integrate with respect to \( x \):\[ \int dy = \int \left(\frac{2}{x^3} - x^3\right) dx \]

Step 4: Perform the Integration

Calculate the left integral as \( y \) and split the right integral:\[ y = \int \frac{2}{x^3} \, dx - \int x^3 \, dx \]

Step 5: Integrate Each Term

Find the antiderivative of each term:- \( \int \frac{2}{x^3} \, dx = \int 2x^{-3} \, dx = 2 \cdot \left( -\frac{1}{2x^2} \right) = -\frac{1}{x^2} \)- \( \int x^3 \, dx = \frac{x^4}{4} \)

Step 6: Combine the Integrals

Add the results from the individual integrations and include the constant of integration \( C \):\[ y = -\frac{1}{x^2} - \frac{x^4}{4} + C \]

Step 7: Present the General Solution

The general solution to the differential equation is:\[ y(x) = -\frac{1}{x^2} - \frac{x^4}{4} + C \], where \( C \) is the constant of integration.

Key Concepts

First-Order Differential Equations

A first-order differential equation is an equation that involves the derivative of a function. In simpler terms, it relates a function and its first derivative. These equations are important because they describe rates of change in various scenarios, such as physics, engineering, and economics.
For the given problem, the equation is \( \frac{dy}{dx} = \frac{2}{x^3} - x^3 \). This presents a relationship between \( y \) and its rate of change with respect to \( x \).
To solve these equations, we need to identify the type of first-order differential equation we have. In this case, it's a linear, first-order, and separable differential equation, allowing us the use of various methods to find its solution.

Separable Differential Equations

Separable differential equations allow us to separate variables and solve using integration. This method involves rearranging the equation so that all terms involving the dependent variable \( y \) are on one side of the equation and all terms with the independent variable \( x \) are on the other.
By transforming the given equation into \( dy = (\frac{2}{x^3} - x^3) \, dx \), we can see that the variables are now separable. This is key because it allows us to integrate both sides independently.

• Step 1: Differentiate the variables: \( dy \) and \( dx \).
• Step 2: Integrate each side separately to solve for \( y \).

Separable equations are quite common and allow for a straightforward approach to finding general solutions.

Integration Techniques

Solving the equation requires integration, which is finding the antiderivative of each term. For this problem, the specific techniques used include integrating simple power functions and applying basic rules.
Consider the expression to integrate \( \int (\frac{2}{x^3} - x^3) \ dx \). This requires splitting into two separate integrals.

• The first term \( \int \frac{2}{x^3} \, dx = \int 2x^{-3} \, dx = -\frac{1}{x^2} \).
• The second term \( \int x^3 \, dx = \frac{x^4}{4} \).

Remember to add the constant of integration \( C \) at the end. Integration techniques help solve the general solution \( y(x) = -\frac{1}{x^2} - \frac{x^4}{4} + C \), reflecting all possible solutions of the differential equation.