Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

First find the general solution (involving a constant \(\mathrm{C}\) ) for the given differential equation. Then find the particular solution that satisfies the indicated condition. $$ \frac{d y}{d x}=\sqrt{\frac{x}{y}} ; y=4 \text { at } x=1 $$

Step-by-Step Solution

Answer

The particular solution is \(y^{3/2} = x^{3/2} + 7\).

Step 1: Separate Variables

To solve the differential equation \(\frac{dy}{dx} = \sqrt{\frac{x}{y}}\), start by separating the variables. Rewrite it as \(y^{1/2} dy = x^{1/2} dx\). This allows us to integrate each side separately.

Step 2: Integrate Both Sides

Integrate both sides: \(\int y^{1/2} dy = \int x^{1/2} dx\). The left integral is \(\frac{2}{3}y^{3/2}\) and the right is \(\frac{2}{3}x^{3/2}\). So, \(\frac{2}{3}y^{3/2} = \frac{2}{3}x^{3/2} + C\), where \(C\) is the integration constant.

Step 3: Simplify the General Solution

Multiply the entire equation by \(3/2\) to simplify: \(y^{3/2} = x^{3/2} + C'\), where \(C' = \frac{3}{2}C\). This expression represents the general solution.

Step 4: Apply the Initial Condition

Use the condition \(y = 4\) when \(x = 1\) to find \(C'\). Replace \(y\) with 4 and \(x\) with 1 in the general solution: \(4^{3/2} = 1^{3/2} + C'\). Calculate \(4^{3/2} = 8\), so \(8 = 1 + C'\). Thus, \(C' = 7\).

Step 5: Write the Particular Solution

Substitute \(C' = 7\) back into the general solution to obtain the particular solution: \(y^{3/2} = x^{3/2} + 7\). This is the specific solution satisfying the initial condition.

Key Concepts

Separation of Variables

Separation of variables is an essential method for solving differential equations. It involves rearranging the equation to isolate the variables on different sides, allowing us to integrate each one on its own. In our problem, the differential equation is given as \(\frac{dy}{dx} = \sqrt{\frac{x}{y}}\). We achieve separation by cross-multiplying to get \(y^{1/2} dy = x^{1/2} dx\).

• This transformation places all terms involving \(y\) on one side.
• Likewise, all terms involving \(x\) go to the other side.

By separating variables, each side can be integrated independently.

Integration

Integration is the next step after separating variables. It helps us find a function whose derivative gives the original equation. In our exercise, we have \(\int y^{1/2} dy = \int x^{1/2} dx\).

• The left side integrates to \(\frac{2}{3}y^{3/2}\).
• The right side becomes \(\frac{2}{3}x^{3/2}\).

After integration, we add an arbitrary constant \(C\) to the equation, forming a new expression as part of the general solution: \(\frac{2}{3}y^{3/2} = \frac{2}{3}x^{3/2} + C\).Simplifying this helps in further steps, revealing the underlying pattern and dependencies between \(x\) and \(y\).

Initial Conditions

Initial conditions are specific values given in a problem, which allow us to determine the value of the constant in the general solution. Given the condition \(y = 4\) at \(x = 1\), we can substitute these values into our simplified general equation. By doing so:

• Calculate \(4^{3/2} = 8\).
• Since \(1^{3/2} = 1\), substituting gives \(8 = 1 + C'\), leading to \(C' = 7\).

This process ensures that the constant means the solution will satisfy the initial condition provided.

Particular Solution

The particular solution is derived from the general solution by using the calculated constant from the initial conditions. In conclusion, substituting \(C' = 7\) into the general solution \(y^{3/2} = x^{3/2} + C'\), yields \(y^{3/2} = x^{3/2} + 7\).

• This solution is tailored to meet specific criteria.
• It verifies that our model accurately describes the scenario at the specified points.

Such particular solutions are crucial, as they offer the specific functions we are seeking, incorporating all the problem's conditions.