Question

Which of the following transformation reduce the differential equation $\frac{dz}{dx} + \frac{z}{x}$log z =$\frac{z}{x^{2}}$ (log z)² into the form $\frac{du}{dx}$ + P

(x) u = Q (x)?

• A. u = log z
• B. u = e^z
• C. u = (log z)⁻¹
• D. u = (log z)²

Answer 1

Answer: (C)

u = (log z)⁻¹

Answer 2

Dividing the given equation by z (log z)², we get

$\frac{1}{z(\log z)^{2}}\frac{dz}{dx} + \frac{1}{\log z}\frac{1}{x} = \frac{1}{x^{2}}$. … (1)

Writing $\frac{1}{\log z}$ = u, we have $\frac{du}{dx}$ = − (log z)⁻² $\frac{1}{z}\frac{dz}{dx}$.

So (1) can be written as

−$\frac{du}{dx} + \frac{u}{x} = \frac{1}{x^{2}}$ ⇒ $\frac{du}{dx} - \frac{u}{x} = \frac{- 1}{x^{2}}$

which is the required form with P (x) = $\frac{- 1}{x}$ and Q (x) = $\frac{- 1}{x^{2}}$