Class 12 Differential Equations · Long Answer3 Markshomogeneous_differential_equations

Class 12 Maths · Differential Equations · Long Answer #50

Solve x dy − y dx − √(x² + y²)dx = 0.

✅ Step-by-Step Solution

Final Answer: y = (Cx² − 1/C)/2, C > 0, x ∈ ℝ.

For x > 0 put v = y/x. The equation gives xv′ = √(1 + v²), so asinh v = ln x + A.
Writing C = eᴬ > 0 gives v = (Cx − 1/(Cx))/2 and hence y = (Cx² − 1/C)/2.
For this expression √(x² + y²) = (Cx² + 1/C)/2, and xy′ − y = (Cx² + 1/C)/2. Thus it also solves the equation for x < 0 and at x = 0.
The family is y = (Cx² − 1/C)/2, C > 0. Equivalently y + √(x² + y²) = Cx² for x ≠ 0, with the smooth extension y(0) = −1/(2C).

[0.5 marks] For x > 0 put v = y/x. The equation gives xv′ = √(1 + v²), so asinh v = ln x + A.
[1 mark] Writing C = eᴬ > 0 gives v = (Cx − 1/(Cx))/2 and hence y = (Cx² − 1/C)/2.
[1 mark] For this expression √(x² + y²) = (Cx² + 1/C)/2, and xy′ − y = (Cx² + 1/C)/2. Thus it also solves the equation for x < 0 and at x = 0.
[0.5 marks] The family is y = (Cx² − 1/C)/2, C > 0. Equivalently y + √(x² + y²) = Cx² for x ≠ 0, with the smooth extension y(0) = −1/(2C).

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