1. The problem statement, all variables and given/known data Find the solution to: y' = x.y.cos(x^2) 2. Relevant equations Integration by Parts method. 3. The attempt at a solution Step 1 (dy/dx).(1/y) = x.cos(x2) (1/y) dy = x.cos(x2) dx Step 2 Integrate both sides. ln|y| = integratal of [ x.cos(x2) dx ] Step 3 Using integration by parts... u = cos(x2) => du = -2x.sin(2^x) dv = x => v = 1.dx Step 4 Subbing back in... ln|y| = u.v - integral of v.du = cos(x2) - integral of [-2x.sin(x2)] ln(y) = cos(x2) + 2*integral of [x.sin(x2)] Step 5 Using integration by parts a second time... u = sin(x2) => du = 2x.cos(x2) dv = x => v = 1.dx Step 6 Subbing back in... ln|y| = cos(x2) + 2 ( sin(x2) - 2*integral of x.cos(x2).dx ) Step 7 I stop my attempt there because it just seems to eventually I get to a point where it becomes a function of itself? (ie. the x.cos(x2) What do I do next? or have I gone wrong somewhere? EDIT --- SOME FURTHER WORK, IS THIS CORRECT? Ok so I noticed that we get "integral of [x.cos(x2)]" back in our formula, and up above in Step 2 I declared it to be = ln|y| So I sub ln|y| into the formula and get... ln|y| = cos(x2) + 2sin(x2) - 4*ln|y| 5*ln|y| = cos(x2) + 2sin(x2) ln|y| = (1/5)*(cos(x2) + 2sin(x2)) y = e^(above line) Ta da? lemmy know if this correct please!