Class 12 Differential Equations · Very Long Answer5 Markshomogeneous_differential_equations

Class 12 Maths · Differential Equations · Very Long Answer #71

Solve x dy − y dx = √(x² + y²)dx.

✅ Step-by-Step Solution

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

For x > 0 put v = y/x, so xy′ − y = x²v′.
Then x²v′ = x√(1 + v²), and dv/√(1 + v²) = dx/x.
Integrate asinh v = ln x + A. Set C = eᴬ > 0, so v = (Cx − 1/(Cx))/2.
Hence y = (Cx² − 1/C)/2. Its derivative is Cx, and √(x² + y²) = (Cx² + 1/C)/2.
Substitution gives xy′ − y = √(x² + y²) for every real x. This family also covers x < 0 branches and extends through x = 0 with y(0) < 0.

[1 mark] For x > 0 put v = y/x, so xy′ − y = x²v′.
[1 mark] Then x²v′ = x√(1 + v²), and dv/√(1 + v²) = dx/x.
[1 mark] Integrate asinh v = ln x + A. Set C = eᴬ > 0, so v = (Cx − 1/(Cx))/2.
[1 mark] Hence y = (Cx² − 1/C)/2. Its derivative is Cx, and √(x² + y²) = (Cx² + 1/C)/2.
[1 mark] Substitution gives xy′ − y = √(x² + y²) for every real x. This family also covers x < 0 branches and extends through x = 0 with y(0) < 0.

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