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Given the differential equation x*sin(y/x)*dy = [y*sin(y/x) - x]*dx with initial condition y(1) = pi/2, find cos(y/x).
- x
- 1/x
- log(x)
- e^x
Correct answer: log(x)
Solution
Rewrite: dy/dx = [y*sin(y/x) - x] / [x*sin(y/x)]. Let v = y/x, y = vx, dy/dx = v + x*dv/dx. Substituting: v + x*dv/dx = (vx*sin(v) - x)/(x*sin(v)) = v - 1/sin(v). So x*dv/dx = -1/sin(v), i.e. sin(v)*dv = -dx/x. Integrating: -cos(v) = -ln|x| + C, i.e. cos(y/x) = ln(x) + C. Initial condition y(1) = pi/2: cos(pi/2) = ln(1) + C => C = 0. Therefore cos(y/x) = ln(x) = log(x).
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