Class 12 Differential Equations · Long Answer3 Marksfirst_order_differential_equations

Class 12 Maths · Differential Equations · Long Answer #57

Find the particular solution of eˣ√(1 − y²)dx + (y/x)dy = 0 with y(0) = 1.

✅ Step-by-Step Solution

Final Answer: No classical solution of the equation as printed exists at the stated initial point.

The term y/x in the original equation requires x ≠ 0.
The prescribed point (0, 1) is outside that domain; no differentiable solution of the printed equation can satisfy the IVP on an interval containing 0.
For comparison only, multiplying by x yields a different, extended equation xeˣ√(1 − y²) + yy′ = 0. Away from y = ±1 its integral is √(1 − y²) = (x − 1)eˣ + C.
There is no classical solution of the printed IVP. The extended equation, which admits y = 1 among other local possibilities, must not be substituted silently for it.

[0.5 marks] The term y/x in the original equation requires x ≠ 0.
[1 mark] The prescribed point (0, 1) is outside that domain; no differentiable solution of the printed equation can satisfy the IVP on an interval containing 0.
[1 mark] For comparison only, multiplying by x yields a different, extended equation xeˣ√(1 − y²) + yy′ = 0. Away from y = ±1 its integral is √(1 − y²) = (x − 1)eˣ + C.
[0.5 marks] There is no classical solution of the printed IVP. The extended equation, which admits y = 1 among other local possibilities, must not be substituted silently for it.

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