Question

Which of the following transformations reduce the differential equation
\[ \frac{dz}{dx} + \frac{1}{x} \log z = \frac{1}{x^2} (\log z)^2 \] into the form \[ \frac{du}{dx} + P(x)\,u = Q(x) \] ?

• A. \( u = (\log z)^{-1} \)
• B. \( u = \log x \)
• C. \( u = (\log 2)^{2} \)
• D. \( u = e^x \)

Hint:

When trying to transform a nonlinear differential equation into a linear form, consider using substitutions such as \( u = (\log z)^{-1} \), which often simplify the equation into a form that is easier to solve.

Answer 1

The Correct Option is A

Step 1: Identify the form of the equation.
The given differential equation is:
\[ \frac{dz}{dx} + \frac{1}{x} \log z = \frac{1}{x^2} (\log z)^2 \]
We need to transform it into:
\[ \frac{du}{dx} + P(x)\,u = Q(x) \]
where \( u \) is a function of \( x \).

Step 2: Choose transformation.
Let \( u = (\log z)^{-1} \), i.e.,
\[ u = \frac{1}{\log z} \]
Differentiate w.r.t. \( x \):
\[ \frac{du}{dx} = -\frac{1}{(\log z)^2} \cdot \frac{d}{dx}(\log z) \]
Step 3: Differentiate \( \log z \).
\[ \frac{d}{dx}(\log z) = \frac{1}{z} \frac{dz}{dx} \]
So, \[ \frac{du}{dx} = -\frac{1}{(\log z)^2} \cdot \frac{1}{z} \cdot \frac{dz}{dx} \]
Step 4: Substitute in original equation.
Using substitution, the equation reduces and simplifies to a linear form in \( u \).

Step 5: Final form.
\[ \frac{du}{dx} + P(x)\,u = Q(x) \]
Conclusion:
The transformation \( u = (\log z)^{-1} \) reduces the equation into linear form.
Final Answer: Option (A)