Problem 1
Question
Solve the given differential equation. \(\frac{d y}{d x}=\frac{x}{2 y}\)
Step-by-Step Solution
Verified Answer
\( y^2 = \frac{1}{2}x^2 + C \)
1Step 1: Separate the variables
Arrange the equation so that all terms involving \(y\) are on one side and all terms involving \(x\) are on the other side: \( 2y \, dy = x \, dx \)
2Step 2: Integrate both sides
The next step is to integrate both sides with respect to their respective variables: \( \int 2y \, dy = \int x \, dx \). Evaluating the integrals, we get: \( y^2 = \frac{1}{2}x^2 + C \). This is the general solution of the differential equation.
Key Concepts
Separation of VariablesIntegrationGeneral Solution
Separation of Variables
When solving differential equations, one common technique is separation of variables. This method helps us to rearrange a differential equation so that each variable appears on a different side of the equation. Basically, all terms involving the dependent variable should be on one side, while all terms involving the independent variable are on the other side.
In the original exercise, we start with the equation: \(\frac{dy}{dx} = \frac{x}{2y}\).
To separate the variables, we need to multiply both sides by \(2y\) and multiply both sides by \(dx\) to isolate them like this: \(2y \, dy = x \, dx\).
This rearranges the terms such that all the "\(y\)" terms are with "\(dy\)" and the "\(x\)" terms are with "\(dx\)". From here, we can proceed to integrate both sides of the equation.
In the original exercise, we start with the equation: \(\frac{dy}{dx} = \frac{x}{2y}\).
To separate the variables, we need to multiply both sides by \(2y\) and multiply both sides by \(dx\) to isolate them like this: \(2y \, dy = x \, dx\).
This rearranges the terms such that all the "\(y\)" terms are with "\(dy\)" and the "\(x\)" terms are with "\(dx\)". From here, we can proceed to integrate both sides of the equation.
Integration
Integration is the process we use to find the antiderivative or the integral of a function. By integrating both sides of a differential equation, we can find a relationship between the variables once the derivatives are removed.
In our example, after separating the variables, we're left with: \(\int 2y \, dy = \int x \, dx\).
Let's look at each integral separately.
Therefore, after integration, you have the equation: \(y^2 = \frac{1}{2}x^2 + C\), where \(C\) is the constant of integration that arises because we used indefinite integrals.
In our example, after separating the variables, we're left with: \(\int 2y \, dy = \int x \, dx\).
Let's look at each integral separately.
- For the left side, \(\int 2y \, dy\), the integral of \(2y\) with respect to \(y\) results in \(y^2\).
- For the right side, \(\int x \, dx\), the integral of \(x\) with respect to \(x\) is \(\frac{1}{2}x^2\).
Therefore, after integration, you have the equation: \(y^2 = \frac{1}{2}x^2 + C\), where \(C\) is the constant of integration that arises because we used indefinite integrals.
General Solution
Once you've solved the differential equation through separation of variables and integration, the result you get is known as the general solution. This solution includes an arbitrary constant \(C\), representing an infinite family of possible solutions depending on initial conditions or additional information.
In the context of our solved equation, the general solution was obtained as: \(y^2 = \frac{1}{2}x^2 + C\).
This expression shows a relationship between \(x\) and \(y\), and it describes a set of all possible curves that solve the original equation.
In the context of our solved equation, the general solution was obtained as: \(y^2 = \frac{1}{2}x^2 + C\).
This expression shows a relationship between \(x\) and \(y\), and it describes a set of all possible curves that solve the original equation.
- The presence of \(C\) means that we have not yet specified a unique curve; different values of \(C\) correspond to different particular solutions.
- To find a unique solution, more information, such as initial conditions, would be needed.
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