Solve this exact differential equation: y + x dy / dx = 0 y = c/x You can identi

Question

Solve this exact differential equation: y + x dy / dx = 0 y = c/x You can identify that M(x, y) = y or partialf (x, y) / partialx = y Integrate to get f(x, y) = xy + g(y) where g(y) is a function. Do the same with the second part of the equation. Identify that N(x, y) = x Because partialf(x, y) / partialy = N(x, y) that means partial f(x, y) / partial y = x If partialf(x, y) / partial y = x + partialg(y) / partialy then partialg(y) / partialy = 0 Integrate again to get g(y) = k where k is a constant of integration. Because f(x, y) = xy + g(y) you get f(x, y) = xy + k In general, the solution is f(x, y) = c so xy + k = c Absorbing k into c gives you y = c/x

Explanation / Answer

Solve for ( dy(x))/( dx): ( dy(x))/( dx) = -(y(x))/x Divide both sides by y(x): (( dy(x))/( dx))/(y(x)) = -1/x Integrate both sides with respect to x: integral (( dy(x))/( dx))/(y(x)) dx = integral -1/x dx Evaluate the integrals: log(y(x)) = -log(x)+c_1, where c_1 is an arbitrary constant. Solve for y(x): y(x) = e^(c_1)/x Simplify the arbitrary constant: y(x) = c_1/x

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