We are given the differential equation $\frac{dy}{dx} = \frac{3xy}{(\ln y)^6}$ and need to find the solution through the point $(x, y) = (0, e)$. We aim to express the solution implicitly in terms of $\ln y$.

Step 1: Separate the Variables

We start by separating the variables $x$ and $y$:

$\frac{dy}{y} = \frac{3x \, dx}{(\ln y)^6}$

Step 2: Integrate Both Sides

Next, integrate both sides with respect to their respective variables:

$\int \frac{dy}{y} = \int \frac{3x \, dx}{(\ln y)^6}$

Step 3: Solve the Left Side

The left side integrates as:

$\int \frac{dy}{y} = \ln |y| + C_1$

However, given the initial condition that $y = e$ (which is positive), we can simply write:

$\ln y = \ln y + C_1$

Step 4: Solve the Right Side

The right side needs integration. However, the right side still has $\ln y$ in it, so we notice that we might need a different approach to integrate effectively. Instead, notice that:

$\frac{dy}{y} = \frac{3x \, dx}{(\ln y)^6}$

suggests a substitution. Let $u = \ln y$. Then $du = \frac{1}{y} dy$, so the equation becomes:

$du = \frac{3x \, dx}{u^6}$

Now integrate both sides:

$\int u^6 \, du = \int 3x \, dx$

This simplifies to:

$\frac{u^7}{7} = \frac{3x^2}{2} + C_2$

Substitute back $u = \ln y$:

$\frac{(\ln y)^7}{7} = \frac{3x^2}{2} + C_2$

Step 5: Apply Initial Condition

We use the initial condition $(x, y) = (0, e)$:

$\frac{(\ln e)^7}{7} = \frac{3 \times 0^2}{2} + C_2$

Since $\ln e = 1$:

$\frac{1^7}{7} = C_2$

So:

$C_2 = \frac{1}{7}$

Step 6: Write the Final Implicit Solution

Finally, the implicit solution is:

$\frac{(\ln y)^7}{7} = \frac{3x^2}{2} + \frac{1}{7}$

Or equivalently:

$(\ln y)^7 = \frac{21x^2 + 2}{2}$

This is the implicit solution for $\ln y$ in terms of $x$.

Would you like further details, or do you have any questions?

5 Related Questions

1. How do you solve separable differential equations?
2. What are the common methods for solving first-order differential equations?
3. How does the initial condition affect the integration constant in differential equations?
4. What is the significance of an implicit solution versus an explicit one?
5. Can we always solve differential equations using substitution?

Tip

When solving differential equations, always check if the equation can be simplified by substitution or separation of variables before attempting more complex methods.