Key Concepts and Formulas

• Separation of Variables: A method for solving first-order differential equations by isolating terms involving the dependent variable ($y$) and its differential ($dy$) on one side and terms involving the independent variable ($x$) and its differential ($dx$) on the other. The goal is to obtain the form $g(y)dy = h(x)dx$.
• Integration of $\frac{f'(x)}{f(x)}$: $\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C$, where $C$ is the constant of integration.
• Derivative of Exponential Function: $\frac{d}{dx}(a^x) = a^x \ln a$.

Step-by-Step Solution

1. Analyze and Simplify the Given Differential Equation

We are given the differential equation: $\frac{dy}{dx} = \frac{2^x y + 2^y \cdot 2^x}{2^x + 2^{x+y} \log_e 2}$ Our first step is to simplify the expression on the right-hand side by factoring.

• Simplifying the Numerator: Factor out $2^x$ from the numerator: $2^x y + 2^y \cdot 2^x = 2^x (y + 2^y)$

• Simplifying the Denominator: Rewrite $2^{x+y}$ as $2^x \cdot 2^y$ and factor out $2^x$ from the denominator: $2^x + 2^{x+y} \log_e 2 = 2^x + 2^x \cdot 2^y \log_e 2 = 2^x (1 + 2^y \log_e 2)$

• Substituting Back into the ODE: Substitute these simplified expressions back into the original differential equation: $\frac{dy}{dx} = \frac{2^x (y + 2^y)}{2^x (1 + 2^y \log_e 2)}$ Cancel out the $2^x$ term from both the numerator and the denominator: $\frac{dy}{dx} = \frac{y + 2^y}{1 + 2^y \log_e 2}$

2. Separate the Variables

Rearrange the equation to isolate $y$ terms with $dy$ and $x$ terms with $dx$: $\frac{dy}{dx} = \frac{y + 2^y}{1 + 2^y \log_e 2}$ Multiply both sides by $(1 + 2^y \log_e 2)$ and by $dx$: $(1 + 2^y \log_e 2) dy = (y + 2^y) dx$ Divide both sides by $(y + 2^y)$: $\frac{1 + 2^y \log_e 2}{y + 2^y} dy = dx$

3. Integrate Both Sides

Integrate both sides of the separated equation with respect to their respective variables: $\int \frac{1 + 2^y \log_e 2}{y + 2^y} dy = \int dx$

• Integrating the Right-Hand Side (RHS): $\int dx = x + C$ where $C$ is the constant of integration.

• Integrating the Left-Hand Side (LHS): The integral on the left side is $\int \frac{1 + 2^y \log_e 2}{y + 2^y} dy$. Let $f(y) = y + 2^y$. Then, $f'(y) = \frac{d}{dy}(y + 2^y) = 1 + 2^y \log_e 2$. The integral is of the form $\int \frac{f'(y)}{f(y)} dy = \ln |f(y)| + C$. Thus, $\int \frac{1 + 2^y \log_e 2}{y + 2^y} dy = \ln |y + 2^y|$

• Combining the Integrals: Equating the results from both sides, we get the general solution to the differential equation: $\ln |y + 2^y| = x + C$

4. Use the Initial Condition to Find the Constant of Integration (C)

We are given the initial condition $y(0) = 0$. Substitute $x=0$ and $y=0$ into the general solution: $\ln |0 + 2^0| = 0 + C$ $\ln |1| = C$ $0 = C$ Thus, $C = 0$. Substitute $C=0$ back into the general solution: $\ln |y + 2^y| = x$ Since we will be considering $y=1$, we can drop the absolute value sign: $\ln (y + 2^y) = x$

5. Find the Value of x for y = 1

Substitute $y=1$ into the particular solution: $x = \ln (1 + 2^1)$ $x = \ln (1 + 2)$ $x = \ln 3$

6. Determine the Interval for x

We have $x = \ln 3$. We know that $e \approx 2.718$ and $e^2 \approx 7.389$. Since $e < 3 < e^2$, then $\ln e < \ln 3 < \ln e^2$, which implies $1 < \ln 3 < 2$. Therefore, $x = \ln 3$ lies in the interval $(1, 2)$.

Common Mistakes & Tips:

• Simplification First: Always simplify the differential equation before separating variables.
• Recognize Integral Forms: Be familiar with common integral forms, such as $\int \frac{f'(x)}{f(x)} dx = \ln|f(x)| + C$.
• Constant of Integration: Do not forget the constant of integration, $C$.

Summary:

We solved the differential equation by separation of variables, simplified the equation by factoring, integrated both sides (recognizing the form of the integral on the left-hand side), used the initial condition to find the constant of integration, and found the value of $x$ when $y=1$. Finally, we determined the interval for $x$ by comparing $\ln 3$ to $\ln e$ and $\ln e^2$.

The final answer is \boxed{(1, 2)}, which corresponds to option (A).