The general solution of the differential equation x(1 + y^2)dx + y (1 + x^2)dy = 0 is (1 + x^2) (1 + y^2) = k.
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The given differential equation is
x (1 + y²) dx + y (1 + x²) dy = 0
⇒ (x dx) / (1 + x²) + (y dy) / (1 + y²) = 0
⇒ (2x dx) / (1 + x²) + (2y dy) / (1 + y²) = 0
On integration, we get
∫ (2x dx) / (1 + x²) + ∫ (2y dy) / (1 + y²) = 0
⇒ log(1 + x²) + log(1 + y²) = logk, where logk is integral constant
⇒ log{(1 + x²) (1 + y²)} = logk
⇒ (1 + x²) (1+ y²) = k,
which is the required general solution.
Hence, proved.
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The given differential equation is
x (1 + y²) dx + y (1 + x²) dy = 0
⇒ (x dx) / (1 + x²) + (y dy) / (1 + y²) = 0
⇒ (2x dx) / (1 + x²) + (2y dy) / (1 + y²) = 0
On integration, we get
∫ (2x dx) / (1 + x²) + ∫ (2y dy) / (1 + y²) = 0
⇒ log(1 + x²) + log(1 + y²) = logk, where logk is integral constant
⇒ log{(1 + x²) (1 + y²)} = logk
⇒ (1 + x²) (1+ y²) = k,
which is the required general solution.
Hence, proved.
#MarkAsBrainliest
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