Question

please give me its solution

Answer 1

$\bold{Answer :}$

Given that,

y' = 1 + x + y² + xy²

→ dy/dx = 1 + x + y² (1 + x)

→ dy/dx = (1 + x) (1 + y²)

→ dy/(1 + y²) = dx/(1 + x)

Now, taking integration, we get

∫ dy/(1 + y²) = ∫ dx/(1 + x)

→ tan⁻¹ (y) = log (1 + x) + c ...(i), where c is integral constant

Given that, y (0) = 0

When x = 0, y = 0

From the above equation (i), we get

tan⁻¹ (0) = log (1 + 0) + c

→ c = 0

From (i), we get the required solution as

tan⁻¹ (y) = log (1 + x)

→ y = tan {log (1 + x)}

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