NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.4 Question 7

✅ Step-by-Step Solution

1. After division the slope depends only on y/x (on a fixed sign component when radicals occur). Put y=vx, so y′=v+xv′.
2. The reduced equation is x dv/dx=\(\frac{2 v \cos{\left(v \right)}}{v \sin{\left(v \right)} - \cos{\left(v \right)}}\). Separate dv/(\(\frac{2 v \cos{\left(v \right)}}{v \sin{\left(v \right)} - \cos{\left(v \right)}}\))=dx/x on branches where division is valid.
3. Integration and back-substitution give \(x y \cos{\left(\frac{y}{x} \right)}\) = C.
4. Integrating the reduced equation uses −½log|v cos v|=log|x|+C. Clearing the logarithm gives xy cos(y/x)=C. The zero-constant straight branches are retained.
5. Differentiate the resulting relation to recover the equation on intervals where the original expressions and the required slope are defined.