NCERT Class 12 Maths · Chapter 9 Differential Equations · Exercise 9.4 Question 15

Question

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

Accepted Answer

Final Answer: y=2x/(1−log x), 0<x<e. 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=$\frac{v^{2}}{2}$. Separate dv/($\frac{v^{2}}{2}$)=dx/x on branches where division is valid. Step 3: Integration and back-substitution give $- \frac{2 x}{y} - \log{\left(\left|{x}\right| \right)}$ = C. Step 4: At (1,2), C=−1. Solve for y to obtain y=2x/(1−log x). The initial interval is 0<x<e. 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 15

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