Let a curve $y = f(x)$ pass through the points $(0, 5)$ and $(\log_e 2, k)$. If the curve satisfies the differential equation $2(3 + y) e^{2x} dx - (7 + e^{2x}) dy = 0$, then $k$ is equal to

Step 1: Separate the variables

The given differential equation is $2(3 + y) e^{2x} dx - (7 + e^{2x}) dy = 0$. To solve this, we need to separate the variables $x$ and $y$. We rearrange the equation to group all terms involving $y$ with $dy$ and all terms involving $x$ with $dx$. This gives us a separable differential equation.

Step 2: Integrate both sides

Now that the variables are separated, we integrate both sides of the equation. The integral of $\frac{1}{3+y}$ with respect to $y$ is $\log|3+y|$. For the right side, we can use a substitution $u = 7+e^{2x}$, which means $du = 2e^{2x} dx$. So the integral becomes $\int \frac{1}{u} du = \log|u|$.

Step 3: Solve the integrals

After integrating both sides, we get $\log|3+y| = \log|7+e^{2x}| + C$, where $C$ is the constant of integration. We can rewrite the constant $C$ as $\log A$ for some constant $A > 0$.

Step 4: Simplify the general solution

Using the logarithm property $\log a + \log b = \log(ab)$, we can combine the terms on the right side: $\log|7+e^{2x}| + \log A = \log|A(7+e^{2x})|$. Then, by exponentiating both sides, we remove the logarithms, yielding $3+y = A(7+e^{2x})$.

Step 5: Use the initial condition to find A

The curve passes through the point $(0, 5)$. We substitute $x=0$ and $y=5$ into the general solution to find the value of the constant $A$. This gives $3+5 = A(7+e^{0})$, which simplifies to $8 = A(7+1)$, so $8 = 8A$, meaning $A=1$.

Step 6: Find k using the second point

Now that we have the specific solution $3+y = 7+e^{2x}$, we use the second point $(\log_e 2, k)$. Substitute $x=\log_e 2$ and $y=k$ into the equation. We use the logarithm property $e^{\log_e M} = M$ and $a\log b = \log b^a$. So $e^{2\log_e 2} = e^{\log_e 2^2} = e^{\log_e 4} = 4$. This leads to $3+k = 7+4$, which simplifies to $3+k=11$, and thus $k=8$.

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