Question

Obtain the general solution to the equation. (x²+3) dy dx + xy - 7x = 0 The general solution is y(x) = ignoring lost solutions, if any.

Answer 1

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Task: solve (x² + 3) y'(X) + x y(X) – 7 x = 0 Answer: y(x) = C1 +7 V x2 +3 Ordinary differential equation classification: first-order linear ordinary differential equation Step by step solution: dy(x) Solve the separable equation - 7 x + (x2 + 3) + x y(x) = 0 dx dy(x) Solve for dx dy(x) 7 x - x y(x) dx x² + 3 Simplify: dy(x) X(-y(x) + 7) dx x2 + 3 Divide both sides by - y(x) + 7: dyx) dx -y(x) + 7 x2 + 3 Integrate both sides with respect to x: dy(x) S dx= Sri - y(x)+7 dx x2 + 3 Evaluate the integrals: -log(-y(x) + 7) = log(x + 3) +C1, where cy is an arbitrary constant. 2 1 Solve for y(x): e 1 y(x) = - -7 V x2 + 3 Simplify the arbitrary constants: Answer: C1 y(x) = + 7 V x2 + 3

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