Question

Find the general solution of each of the following differential equations. \(2 x y^{\prime}+y=2 x^{5 / 2}\)

Short answer

Simplify each term with respect to x and y to differentiate both sides of the expression.

Step-by-step solution

Rewrite the Differential Equation

First, rewrite the given differential equation in a more standard form. The given differential equation is: \[2 x y^{\text{'} }+ y = 2 x^{5/2} \]

Isolate the Derivative

Isolate the term involving the derivative, which is \[2 x y^{\text{'}} \]. To do this, rearrange the equation: \[2 x y^{\text{'}} = 2 x^{5/2} - y\]

Divide by the Coefficient of the Derivative

Divide both sides by the coefficient of the derivative, which is \[2 x \]: \[y^{\text{'} }= x^{3/2} - \frac{y}{2x} \]

Separate Variables

Separate the variables y and x to integrate both sides. This requires manipulating the equation to the form: \[2 x y^{\text{'}}+ y = 2 x^{5/2}, \] which can be separated as follows: \[2 x y^{\text{'}} = 2 x^{5/2} - y\]. Move \[y \] to the other side of the equation: \[2 x y^{\text{'}} + y = 2 x^{5 / 2} \]

Integrate Both Sides

Integrate both sides with respect to \[x \]. First, the left-hand side: \[\frac{d}{dx}(y\frac{1}{x^{-3/2})}=1\]. Next, the right-hand side: \[\frac{d}{dx}(a x^{-5/2})-2=x\]

Simplify

Combine like terms to write the final differential equations.

Key concepts

Integration Techniques

When solving differential equations, understanding various integration techniques is crucial. One common technique is the method of separation of variables, which simplifies the equation into integrable parts.
This allows us to tackle each part individually, making the integration process more straightforward.

Another important technique involves integrating both sides of the equation with respect to a particular variable. In our problem, we integrate with respect to \(x\) to find the function \(y\).
These integration techniques are foundational tools that help transform complex equations into solvable forms.

Separable Equations

A differential equation is considered separable if it can be written in the form \[\frac{dy}{dx} = g(x)h(y)\]. This means the derivative of \(y\) with respect to \(x\) can be expressed as the product of two functions, one purely of \(x\) and the other of \(y\).

In our example, to approach solving the equation \(2 x y^{\text{'}} + y = 2 x^{5/2}\), we first isolate the derivative term and then rearrange the equation into separable parts:

• Initial Rearrangement: \(2 x y^{\text{'}} = 2 x^{5/2} - y\)
• Divide by \(2x\): \(y^{\text{'}} = x^{3/2} - \frac{y}{2x}\)

This form allows us to treat \(x\) and \(y\) independently, simplifying the integration process.

General Solution

The goal in solving a differential equation is to find the general solution. This solution includes all possible specific solutions and often contains an arbitrary constant \(C\).

For our equation \(2 x y^{\text{'}} + y = 2 x^{5/2}\), the journey to the general solution involves isolating the derivative, rearranging terms, and performing integration:

• Rewriting the given equation
• Isolating the derivative term \(y^{\text{'}}\)
• Making the equation separable and integrating both sides

Once integrated, we add the arbitrary constant \(C\) to ensure our general solution represents all possible specific solutions.
This general solution provides a comprehensive set of functions that satisfy the original differential equation.

Step-by-Step Solution

Breaking down a differential equation problem into steps helps clarify the solution process. Let's revisit our example step-by-step:

• Step 1: Rewrite the given equation: \(2 x y^{\text{'} }+ y = 2 x^{5/2} \).
• Step 2: Isolate the term involving the derivative: \(2 x y^{\text{'}} = 2 x^{5/2} - y\).
• Step 3: Divide by \(2x\) to simplify: \(y^{\text{'} }= x^{3/2} - \frac{y}{2x}\).
• Step 4: Separate variables to prepare for integration: \(2 x y^{\text{'}} + y = 2 x^{5 / 2} \).
• Step 5: Integrate both sides with respect to \(x\).
• Step 6: Simplify to find the general solution.

This structured approach not only helps in solving the problem but also in understanding the underlying concepts and techniques.
It is essential to practice these steps to become proficient in solving similar differential equations.