Given Part:

• The differential equation provided is: $\frac{e^x}{x} = \left(\frac{y + 1}{x}\right)^2 y'$
• Initial condition: $y(0) = 2$

To Find Part:

The goal is to find the function $y(x)$ that satisfies the differential equation and the initial condition using the separation of variables method.

Formulas Used:

1. Separation of variables for differential equations: $\frac{dy}{dx} = f(x)g(y) \Rightarrow \int \frac{1}{g(y)} dy = \int f(x) dx$

Concept Used:

The separation of variables method involves rearranging the differential equation so that all terms involving $y$ are on one side of the equation and all terms involving $x$ are on the other. After separation, both sides are integrated to find the solution. The initial condition is then used to find the constant of integration.

Complete Calculation:

1. Rewrite the equation to isolate $y'$: $\frac{e^x}{x} = \left(\frac{y + 1}{x}\right)^2 y'$ Rearrange to isolate $y'$: $y' = \frac{e^x}{x} \cdot \frac{x^2}{(y + 1)^2}$ Simplify: $y' = \frac{e^x x}{(y + 1)^2}$

2. Separate the variables $y$ and $x$: $\frac{(y + 1)^2}{dy} = e^x x dx$

3. Integrate both sides: $\int (y + 1)^{-2} dy = \int e^x x dx$ Left Side Integration: $\int (y + 1)^{-2} dy = \int \frac{1}{(y + 1)^2} dy = -\frac{1}{y + 1}$ Right Side Integration (using Integration by Parts): Let $u = x$ and $dv = e^x dx$. Then $du = dx$ and $v = e^x$. $\int e^x x dx = x e^x - \int e^x dx = x e^x - e^x = e^x(x - 1)$

   So, the equation becomes: $-\frac{1}{y + 1} = e^x (x - 1) + C$

4. Solve for $y$: $\frac{1}{y + 1} = -e^x (x - 1) - C$ Rearrange to find $y$: $y + 1 = \frac{1}{-e^x (x - 1) - C}$ $y = \frac{1}{-e^x (x - 1) - C} - 1$

5. Determine the constant $C$ using the initial condition $y(0) = 2$: Substitute $x = 0$ and $y = 2$: $2 = \frac{1}{-e^0(0 - 1) - C} - 1$ Simplify: $2 = \frac{1}{1 - C} - 1$ $3 = \frac{1}{1 - C}$ Solve for $C$: $1 - C = \frac{1}{3}$ $C = \frac{2}{3}$

6. Write the final solution: Substitute $C = \frac{2}{3}$: $y = \frac{1}{-e^x (x - 1) - \frac{2}{3}} - 1$

Final Answer:

$y = \frac{1}{-e^x (x - 1) - \frac{2}{3}} - 1$

Do you want details or have any questions?

5 Related Questions:

1. What is the purpose of using the separation of variables method in solving differential equations?
2. How can integration by parts help in solving certain integrals during differential equation solutions?
3. What are the limitations of the separation of variables method?
4. Can every first-order differential equation be solved by separation of variables?
5. How can initial conditions be used to determine constants in the solutions of differential equations?

Tip:

Always check the solution by substituting it back into the original differential equation to verify that it satisfies both the equation and any given initial conditions.