3.8 Q-16

Question Statement

Solve the differential equation:

$$y - x \frac{d y}{d x} = 3 \left(1 + x \frac{d y}{d x}\right)$$

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Background and Explanation

This is a first-order linear differential equation. The method used to solve this is separation of variables, where we isolate $y$ terms on one side and $x$ terms on the other side. It requires basic algebraic manipulation and integration. The key concept involves simplifying the equation by isolating $\frac{dy}{dx}$ and then integrating both sides.

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Solution

1. Start with the given equation:

   The equation is:

$$y - x \frac{d y}{d x} = 3 \left(1 + x \frac{d y}{d x}\right)$$

2. Expand the right-hand side:

   Distribute the $3$ on the right-hand side:

$$y - x \frac{d y}{d x} = 3 + 3x \frac{d y}{d x}$$

3. Rearrange to isolate $\frac{dy}{dx}$:

   Bring all terms involving $\frac{dy}{dx}$ to one side and constant terms to the other:

$$y - x \frac{d y}{d x} - 3x \frac{d y}{d x} = 3$$

Simplifying further:

$$y - 4x \frac{d y}{d x} = 3$$

4. Isolate $\frac{dy}{dx}$:

   Move the term involving $\frac{dy}{dx}$ to one side:

$$-4x \frac{d y}{d x} = 3 - y$$

5. Separate the variables:

   Now, separate the variables so that all $y$-terms are on one side and all $x$-terms are on the other:

$$\frac{-4}{3 - y} , dy = \frac{1}{x} , dx$$

This is now a separable equation.

6. Integrate both sides:

   • For the left-hand side, integrate $\frac{4}{y - 3}$:

$$\int \frac{4}{y - 3} , dy = 4 \ln |y - 3|$$

• For the right-hand side, integrate $\frac{1}{x}$:

$$\int \frac{1}{x} , dx = \ln |x|$$

This gives us the equation:

$$4 \ln |y - 3| = \ln |x| + \ln |C|$$

7. Simplify the equation:

   Combine the logarithms on the right-hand side:

$$\ln |(y - 3)^4| = \ln |C| + \ln |x|$$

Exponentiate both sides to get rid of the logarithms:

$$(y - 3)^4 = Cx$$

8. Solve for $y$:

   Finally, solve for $y$:

$$y - 3 = (Cx)^{1/4}$$

Which simplifies to:

$$y = 3 + Cx^{1/4}$$

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Key Formulas or Methods Used

• Separation of Variables: The process of rearranging the equation so that all terms involving $y$ are on one side and all terms involving $x$ are on the other.
• Integration: Standard integrals for $\frac{1}{x}$ and $\frac{1}{y - 3}$.
• Logarithmic Properties: Using logarithmic properties to combine and simplify terms.

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Summary of Steps

1. Start with the equation: $y - x \frac{d y}{d x} = 3 \left(1 + x \frac{d y}{d x}\right)$.
2. Expand the right-hand side: $y - x \frac{d y}{d x} = 3 + 3x \frac{d y}{d x}$.
3. Rearrange to isolate $\frac{dy}{dx}$: $y - 4x \frac{d y}{d x} = 3$.
4. Isolate $\frac{dy}{dx}$: $-4x \frac{d y}{d x} = 3 - y$.
5. Separate the variables: $\frac{-4}{3 - y} , dy = \frac{1}{x} , dx$.
6. Integrate both sides: $4 \ln |y - 3| = \ln |x| + \ln |C|$.
7. Simplify and solve: $(y - 3)^4 = Cx$, and $y = 3 + Cx^{1/4}$.