Question

Find the "general solution" (that is, a solution containing an arbitrary constant) of each of the following differential equations, by separation of variables. Then find a particular solution of each equation satisfying the given boundary conditions. \(x y^{\prime}=y\), \(y=3\) when \(x=2\).

Short answer

The general solution is \(y = Cx\). The particular solution is \(y = \frac{3}{2}x\).

Step-by-step solution

- Rewrite the equation

Rewrite the given differential equation by separating the variables on each side. The original equation is: \[ x y^{\prime} = y \]Rewrite it as: \[ \frac{dy}{y} = \frac{dx}{x} \]

- Integrate both sides

Integrate both sides of the equation separately:\[ \int \frac{1}{y} \, dy = \int \frac{1}{x} \, dx \]The antiderivatives are:\[ \ln|y| = \ln|x| + C \]where \(C\) is the constant of integration.

- Solve for y

Exponentiate both sides to remove the natural logarithms:\[ e^{\ln|y|} = e^{\ln|x| + C} \]This simplifies to:\[ |y| = e^C \, |x| \]Let \(e^C = C^*\) (a new constant), so:\[ y = C^* x \]Thus, the general solution is:\[ y = Cx \]

- Apply the boundary condition

To find the particular solution, use the boundary condition \(y = 3\) when \(x = 2\). Substitute these values into the general solution:\[ 3 = C \, (2) \]Solving for \(C\) gives:\[ C = \frac{3}{2} \]Thus, the particular solution is:\[ y = \frac{3}{2} \, x \]

Key concepts

Separation of Variables

The method of Separation of Variables is a key technique used to solve differential equations. It involves rearranging the equation so that each variable and its derivative are on opposite sides. This means we can integrate both sides independently.
For example, starting with the equation: \[x y' = y\],
We rearrange to separate the variables: \[\frac{dy}{y} = \frac{dx}{x}\].

By doing this, we are able to tackle each part separately, allowing for easier integration and ultimately leading us to the solution.

Integration Techniques

Once the variables are separated, integration techniques are employed to find the antiderivatives. For the equation: \[\int \frac{1}{y} \, dy = \int \frac{1}{x} \, dx\]
We integrate each side separately:

• The antiderivative of \(\frac{1}{y}\) with respect to \(y\) is \(\ln|y|\)
• The antiderivative of \(\frac{1}{x}\) with respect to \(x\) is \(\ln|x|\)

Thus, we get: \[\ln|y| = \ln|x| + C\]
where \(C\) is the constant of integration.

This step is important as it simplifies the relationship between variables, making it easier to find the general solution.

Boundary Conditions

Boundary conditions are essential for finding the particular solution of a differential equation. They provide specific values that the solution must satisfy. In our example, the boundary condition is \(y = 3\) when \(x = 2\).
By substituting these values into the general solution:
\[y = Cx\]
We get: \[3 = C(2)\],
Solving for \(C\) gives:
\[C = \frac{3}{2}\]

This process ensures that our solution fits the given condition, providing an accurate and tailored result.

Particular Solution

After applying the boundary conditions, we derive the particular solution from the general solution. Starting with the general solution: \[y = Cx\],
The boundary condition \(y = 3\) when \(x = 2\) helps us find the specific value for \(C\).

Using \(C = \frac{3}{2}\), we substitute it back into the general solution to get:
\[y = \frac{3}{2}x\]

Therefore, the particular solution, which satisfies the initial condition, is: \[y = \frac{3}{2}x\].
This step customizes the general solution to fit the given problem precisely.