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Solve the first order linear D.E. (y +1)Cosxdx = dy where the G.S. is given y e^∫p(x)dx = ∫Q(x) e^∫p(x)dx  + C

zzzzz  Aug 4, 2017
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Solve the separable equation ( dy(x))/( dx) = (y(x) + 1) cos(x):
Divide both sides by y(x) + 1:
(( dy(x))/( dx))/(y(x) + 1) = cos(x)
Integrate both sides with respect to x:
 integral(( dy(x))/( dx))/(y(x) + 1) dx = integral cos(x) dx
Evaluate the integrals:
log(y(x) + 1) = sin(x) + c_1, where c_1 is an arbitrary constant.
Solve for y(x):
y(x) = e^(sin(x) + c_1) - 1
Simplify the arbitrary constants:
Answer: | y(x) = c_1 e^(sin(x)) - 1

Guest Aug 4, 2017

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