NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.4 Question 13

Question

Show the equation y′=$- \sin^{2}{\left(\frac{y}{x} \right)} + \frac{y}{x}$ is homogeneous and solve it. Initial condition: y(1)=π/4.

Accepted Answer

Final Answer: cot(y/x)=log x+1 on the initial branch. Step 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′. Step 2: The reduced equation is x dv/dx=$- \sin^{2}{\left(v \right)}$. Separate dv/($- \sin^{2}{\left(v \right)}$)=dx/x on branches where division is valid. Step 3: Integration and back-substitution give $- \log{\left(\left|{x}\right| \right)} + \cot{\left(\frac{y}{x} \right)}$ = C. Step 4: Separate −cosec²v dv=dx/x. At x=1,y=π/4 the constant is 1. Step 5: Differentiate the resulting relation to recover the equation on intervals where the original expressions and the required slope are defined.

NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.4 Question 13

💡 Concept Hints

✅ Step-by-Step Solution

Master Chapter 9: Differential Equations

All Questions in Differential Equations — Exercise 9.4

Related Questions in Differential Equations