Question

Math45-Homework- 08: Problem 6 Previous Problem Problem List Next Problem (1 point) The function yi (x) -cos(5x) is a solution to the differential equation y" 25y 0. Use reduction of order to find another solution y2 to this differential equation help (formulas)

Answer 1

"Given that y_1(x) = cos 5x is one sol of y"" + 25y = 0 let y_2 be the other solution let y = y_1y_2. Then y = y_2 cos 5x dy/dx = -5y_2 sin 5x + cos 5x dy_2/dx d^2y/dx^2 = -g5y_2 cos 5x - 5 sin 5x dy_2/dx -dy_2/dx 5 sin 5x + cos 5x d^2y_2/dx^2 then by (1), -25 y_2 cos 5x - 10 sin x dy_2/dx + cos 5x d^2y_2/dx_2 + 25y_2 cos 5x = 0 or d^2y_2/dx^2 = cos 5x - 10 dy_2/dx sin 5x = 0 let dy_2/dx = v then dv/dx = 10 sin 5x/cos 5x v or integral dv/v = 10 integral sin 5x/cos 5x dx let cos 5x = t -5 sin 5xdx = dt or integral dv/v = -10/5 integral 1/t dt or en v = -2 ln t + ln c v = c/t^2 = c/cos^2 5x dy_2/dx = c/cos^2 5x or integral dy_2 = c integral sec^2 5x dx "
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