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

Question

Show the equation y′=$\frac{- x y - y^{2}}{x^{2}}$ is homogeneous and solve it. Initial condition: y(1)=1.

Accepted Answer

Final Answer: y=2x/(3x²−1), x>1/√3. 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=$- v \left(v + 2\right)$. Separate dv/($- v \left(v + 2\right)$)=dx/x on branches where division is valid. Step 3: Integration and back-substitution give $\frac{x^{2} y}{2 x + y}$ = C. Step 4: The initial condition gives C=1/3. Solve 3x²y=y+2x to get y=2x/(3x²−1). Its maximal initial interval is x>1/√3. 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 12

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