NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.4 Question 8

Question

Show the equation y′=$- \sin{\left(\frac{y}{x} \right)} + \frac{y}{x}$ is homogeneous and solve it.

Accepted Answer

Final Answer: x tan(y/(2x))=C where finite, plus y=nπx for every integer n. 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=$- \sin{\left(v \right)}$. Separate dv/($- \sin{\left(v \right)}$)=dx/x on branches where division is valid. Step 3: Integration and back-substitution give $x \tan{\left(\frac{y}{2 x} \right)}$ = C. Step 4: The reduced equation is x v′=−sin v. Integrate csc v dv=−dx/x. Straight-line branches v=nπ were excluded by division and all must be included. 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 8

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