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DIFFERENTIAL EQUATIONS PRACTICE PROBLEMS: ANSWERS

DIFFERENTIAL EQUATIONS PRACTICE PROBLEMS: ANSWERS1. Find the solution ofy0+2xy=x,withy(0) = is a linear equation. The integrating factor iseR2xdx=ex2. Multiplying through by this, we gety0ex2+2xex2y=xex2(ex2y)0=xex2ex2y=Rxe x2dx=12ex2+Cy=12+Ce in the initial condition givesC= 5/2,soy=12 52e= Find the general solution ofxy0=y (y2/x).A number of substitutions will work here. The simplest isy=ux,soy0=u+ gives a separable equation:x(u+u0x)=ux u2x2x=ux u2xdudxx2= u2xZ u 2du=Z1xdx1u=lnx+Cu=1lnx+Cy=xlnx+ Suppose that the frog populationP(t)of a small lake satisfies the DIFFERENTIAL equationdPdt=kP(200 P).(a) Find the equilibrium solutions .

DIFFERENTIAL EQUATIONS PRACTICE PROBLEMS: ANSWERS 1. Find the solution of y0 +2xy= x,withy(0) = −2. This is a linear equation. The integrating factor is e R ... We could use Laplace methods here, but we’ll use the Doperator again. ... and the two vectors for the eigenvalue λ= −2 are clearly independent (neither is a multiple of the other).

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