Solve the differential equation # (dy)/(dx) + x tan(y+x) = 1# ?

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Answer 1 · 2016-06-04T10:16:18

One solution is #y(x) = x + arcsin(e^{-1/2x^2+C})#

First we make the variable change #z(x) = y(x)-x# and

#(dz)/(dx) = (dy)/(dx)-1#

substituting

#1 + (dz)/(dx) + x tan(z) = 1 equiv (dz)/(dx) + x tan(z) = 0#

grouping

#(dz)/(tan(z))=-x dx equiv d/(dx)(log_e(sin(z)))=-d/dx(1/2x^2)#
#log_e sin(z) = -1/2 x^2+ C->sin(z) = e^{-1/2x^2+C}#

and finally

#z(x) = y(x)-x = arcsin(e^{-1/2x^2+C})#

then

#y(x) = x + arcsin(e^{-1/2x^2+C})#