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To solve the differential equation \( x^2 y' - y = x^5 \) using the Bernoulli method with the substitution \( y = u \cdot v \), we first rewrite the equation in a suitable form. 

Given the choices provided, we can analyze the differential equation and apply the substitution method to find the correct form for \( v \).

Starting with the differential equation:
\[ x^2 y' - y = x^5 \]

Using the substitution \( y = u \cdot v \), we need to find \( v \). Differentiating \( y \) with respect to \( x \) using the product rule:
\[ y' = u'v + uv' \]

Substituting \( y \) and \( y' \) back into the original equation:
\[ x^2 (u'v + uv') - uv = x^5 \]

\[ x^2 u'v + x^2 uv' - uv = x^5 \]

To simplify, we assume \( u = x^n \). Therefore, we substitute \( u = x^n \) into the equation:
\[ x^2 (nx^{n-1}v + x^n v') - x^n v = x^5 \]

After simplifying, we get:
\[ x^2 nx^{n-1}v + x^2 x^n v' - x^n v = x^5 \]

Now, match the powers of \( x \) to determine the correct form for \( v \):
\[ nx^{n+1}v + x^{n+2} v' - x^n v = x^5 \]

Assuming \( v = x^m \), we get:
\[ nx^{n+1}x^m + x^{n+2}mx^{m-1} - x^n x^m = x^5 \]

\[ n x^{n+m+1} + mx^{n+m+1} - x^{n+m} = x^5 \]

To balance the powers of \( x \), set:
\[ n + m + 1 = 5 \]
\[ n + m = 4 \]

The equation must hold true for the powers of \( x \):
\[ n + 1 = 4 \rightarrow n = 3 \]
\[ m + 3 + 1 = 5 \rightarrow m = 1 \]

Thus, \( v = x^1 = x \).

So, the correct form for \( v \) is:
\[ d. \; v = x \]

Would you like more details on this solution or have any questions?

Here are five questions you might want to ask next:
1. How do you solve differential equations using other substitution methods?
2. What is the general approach to solving Bernoulli differential equations?
3. Can you provide more examples of solving differential equations using the Bernoulli method?
4. How do you handle non-linear differential equations?
5. What are the applications of Bernoulli differential equations in real-world problems?

Tip: When solving differential equations, carefully choose the right substitution to simplify the equation for easier integration.

Learn how to solve the differential equation x^2 y' - y = x^5 using the Bernoulli method with step-by-step instructions. This problem involves the application of the substitution method, product rule for differentiation, and power rule. Suitable for advanced undergraduate students studying differential equations.

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