Key Concepts and Formulas

• Substitution Method: A technique to simplify differential equations by introducing a new variable.
• Chain Rule of Differentiation: $\frac{d}{dx}f(g(x)) = f'(g(x)) \cdot g'(x)$
• Integration Formulas: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$, $\int \frac{1}{x} \, dx = \log_e |x| + C$

Step-by-Step Solution

1. Analyze the Given Differential Equation and Rearrange

The given differential equation is: $\frac{dy}{dx} - \frac{y + 3x}{\log_e\left( {y + 3x} \right)} + 3 = 0$ We rearrange the equation to isolate the derivative term: $\frac{dy}{dx} + 3 = \frac{y + 3x}{\log_e\left( {y + 3x} \right)}$

Why this step? Rearranging helps us identify a suitable substitution by highlighting the repeated term $(y+3x)$. The structure of the equation suggests that a substitution involving this term will simplify the equation.

2. Choose a Substitution and Differentiate

Let's choose the substitution: $t = y + 3x$ Differentiate both sides with respect to $x$: $\frac{dt}{dx} = \frac{d}{dx}(y + 3x) = \frac{dy}{dx} + 3$

Why this step? We choose $t = y + 3x$ because it appears repeatedly in the equation. Differentiating the substitution allows us to relate $\frac{dt}{dx}$ to $\frac{dy}{dx}$, enabling us to replace $\frac{dy}{dx}$ in the original equation.

3. Substitute into the Original Differential Equation

Substitute $t = y + 3x$ and $\frac{dt}{dx} = \frac{dy}{dx} + 3$ into the rearranged equation: $\frac{dt}{dx} = \frac{t}{\log_e(t)}$

Why this step? This substitution transforms the original equation into a simpler equation involving only $t$ and $x$, making it easier to solve.

4. Separate Variables

We now have the equation: $\frac{dt}{dx} = \frac{t}{\log_e(t)}$ Separate the variables: $\frac{\log_e(t)}{t} dt = dx$

Why this step? Separating variables allows us to integrate each side of the equation independently. Grouping the $t$ terms with $dt$ and the $x$ terms with $dx$ prepares the equation for integration.

5. Integrate Both Sides

Integrate both sides of the equation: $\int \frac{\log_e(t)}{t} dt = \int dx$ Let $u = \log_e(t)$, so $du = \frac{1}{t} dt$. The integral becomes: $\int u \, du = \int dx$ $\frac{u^2}{2} = x + C$ Substitute back $u = \log_e(t)$: $\frac{(\log_e(t))^2}{2} = x + C$

Why this step? Integration is the inverse operation of differentiation. By integrating both sides, we find the general solution that describes the relationship between $t$ and $x$. The constant of integration $C$ accounts for the family of possible solutions.

6. Substitute Back to Express the Solution in Terms of Original Variables

Substitute back $t = y + 3x$: $\frac{(\log_e(y + 3x))^2}{2} = x + C$ Rearrange the equation to match the given options: $x - \frac{1}{2} (\log_e(y + 3x))^2 = -C$ Since $-C$ is also an arbitrary constant, we can replace it with $C$: $x - \frac{1}{2} (\log_e(y + 3x))^2 = C$

Why this step? The final solution must be expressed in terms of the original variables $x$ and $y$. Substituting back allows us to answer the question posed for the original differential equation.

Common Mistakes & Tips

• Forgetting the Constant of Integration: Always include the constant of integration after performing an indefinite integral.
• Incorrect Substitution: Double-check the substitution and its derivative to ensure they are correct.
• Chain Rule: Remember to apply the chain rule when differentiating composite functions.

Summary

We solved the given differential equation using the substitution method. By identifying the repeating term $(y + 3x)$, we introduced a new variable $t = y + 3x$. This transformed the equation into a separable form, which we then integrated. Finally, we substituted back to express the solution in terms of the original variables $x$ and $y$, obtaining the solution $x - \frac{1}{2} (\log_e(y + 3x))^2 = C$.

Final Answer

The final answer is $\boxed{x - \frac{1}{2}{\left( {{{\log }_e}\left( {y + 3x} \right)} \right)^2} = C}$, which corresponds to option (A).