MAT 275 Test 1 SOLUTIONS, FORM A
1. The differential equation 7 isxy00 −exy0+y3=x−
neither linear nor homogeneous.
Solution: Linearity is ruinied by the term; homogeneity is ruined by the on the right-handy3x−7
side.
2. The order of the differential equation 5 ·( )y000 2+ 8 + 2) isy00 −(sin x)y y
20+√y= ln(x
3
Solution: There is a in the equation but no .y000 y0000
3. For the differential equation dy
dt =− −3(y−2)( ( (y+ 1)2y+ 5)3, the function y t) = 5 is
an unstable equilibrium solution
Solution: If F( 3( 2)( (y) = −y−y+ 1)2y+ 5)3, then F(y)<0when yis a little smaller than ,−5
and F(y)>0when yis a little bigger than .−5
4. [Assume that any indefinite integrals are solvable.] The differential equation y2dy
dx =exy4−sin xcan
be solved using
neither an integrating factor nor separation of variables.
Solution: In standard form, the differential equation is dy
dx −exy2=−sin x
y2, and the y2prevents
the existence of an integrating factor. In the form dy
dx =exy2−sin x
y2, it should be clear that the
right-hand side cannot be factored into .F(x) (G y)
1
5. The slope field of a differential equation ) is shown below. Determine the long-term behaviory0=F(x, y
of the solution that satisfies (0) = 1. (The origin is indicated by ay•sign, and the points one unit away
have been marked with a 1.)
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