NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.4 Question 11

Question

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

Accepted Answer

Final Answer: log(x²+y²)+2tan⁻¹(y/x)=log 2+π/2, 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=$- \frac{v^{2} + 1}{v + 1}$. Separate dv/($- \frac{v^{2} + 1}{v + 1}$)=dx/x on branches where division is valid. Step 3: Integration and back-substitution give $\log{\left(x^{2} + y^{2} \right)} + 2 \operatorname{atan}{\left(\frac{y}{x} \right)}$ = C. Step 4: At (1,1), C=log 2+π/2; use the local arctangent branch containing that point. 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 11

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