NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.4 Question 5

Question

Show the equation y′=$\frac{x^{2} + x y - 2 y^{2}}{x^{2}}$ is homogeneous and solve it.

Accepted Answer

Final Answer: (1/(2√2))log|(x+√2y)/(x−√2y)|=log|x|+C, plus y=±x/√2. 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=$1 - 2 v^{2}$. Separate dv/($1 - 2 v^{2}$)=dx/x on branches where division is valid. Step 3: Integration and back-substitution give $- \log{\left(\left|{x}\right| \right)} + \frac{\sqrt{2} \log{\left(\left|{\frac{1 + \frac{\sqrt{2} y}{x}}{1 - \frac{\sqrt{2} y}{x}}}\right| \right)}}{4}$ = C. Step 4: Division excludes v=±1/√2. Both straight lines y=±x/√2 satisfy the original equation and must be added. 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 5

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