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Solve `(dy)/(dx) + x sin 2y = x^(3) cos^(2) y` |
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Answer» We have on dividing by `cos^(2)y` `sec^(2)y(dy)/(dx) + ((2sin y cos y)/(cos^(2) y))x = x^(3)` `rArr sec^(2)y(dy)/(dx) + (2 tan y) x = x^(3)` Set `tan y = v`, so that `sec^(2) y (dy)/(dx) = (dv)/(dx)` Our equation becomes, `(dv)/(dx) + 2vx = x^(3)` which is linear in v `IF = e^(int 2x dx) = e^(x^(2))` multiplying the equation by If and integrating `ve^(x^(2)) = int e^(x^(2)). x^(3)dx + k`, k n being the constant of integration `= int e^(x^(2)). x^(2).xdx + k` (set `t = x^(2)`) `= (1)/(2) int e^(t). t dt + k` `= (1)/(2)e^(t).(t-1) + k` `rArr - tan y e^(x^(2)) = (1)/(2)e^(x^(2)) (x^(2)-1) + k` `rArr 2 tan y e^(x^(2)) (x^(2)+1) + lambda, lambda` being another constant. `:. 2 tan y = (x^(2) - 1) + lambda e^(-x^(2))`. |
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