Question

Convert the non-linear equation \( xy' + y = x^4 y^3 \) into a linear one using the transformation \( z = y^{-2} \).
The linear equation is ...........

• A. \( \dfrac{dz}{dx} - \dfrac{2z}{x} = -2x^3 \)
• B. \( \dfrac{dz}{dx} + \dfrac{2z}{x} = 2x^3 \)
• C. \( \dfrac{dz}{dx} - \dfrac{z}{2x} = 2x^2 \)
• D. \( \dfrac{dz}{dx} + \dfrac{z}{2x} = -2x^2 \)

Hint:

When a non-linear term like \( y^n \) appears, try using a substitution such as \( z = y^{-n+1} \) or another form to simplify and reduce the equation to a linear differential equation.

Answer 1

The Correct Option is A

Step 1: Given equation: \[ xy' + y = x^4 y^3 \] Step 2: Divide through by \( y^3 \): \[ x \cdot \dfrac{y'}{y^3} + \dfrac{1}{y^2} = x^4 \] Step 3: Use the substitution \( z = y^{-2} ⇒ \dfrac{dz}{dx} = -2y^{-3} \dfrac{dy}{dx} = -2 \dfrac{y'}{y^3} \)
Rewriting: \[ \dfrac{y'}{y^3} = -\dfrac{1}{2} \dfrac{dz}{dx} \] Substitute into the equation: \[ x \left( -\dfrac{1}{2} \dfrac{dz}{dx} \right) + z = x^4 ⇒ -\dfrac{x}{2} \dfrac{dz}{dx} + z = x^4 \] Multiply through by 2: \[ - x \dfrac{dz}{dx} + 2z = 2x^4 ⇒ \dfrac{dz}{dx} - \dfrac{2z}{x} = -2x^3 \] Final Answer: \( \boxed{\dfrac{dz}{dx} - \dfrac{2z}{x} = -2x^3} \)