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Chapter 1: Q8 E (page 14)

Question: In Problems 3鈥�8, determine whether the given function is a solution to the given differential equation.

y=3sin2x+e-x, y''+4y=5e-x

Short Answer

Expert verified

The given function is a solution to the given differential equation.

Step by step solution

01

Differentiating the given equation w.r.t. (with respect to) x

Firstly, we will differentiate y=3sin2x+e-xwith respect to x,

dydx=6cos2x-e-x

Again, differentiatingthe given function with respect to x,

d2ydx2=-12sin2x+e-x

02

Simplification

Putting the values from step 1 in the L.H.S. (Left-hand side) of the given differential equation,

y''+4y=-12sin2x+e-x+43sin2x+e-xy''+4y=-12sin2x+e-x+12sin2x+4e-xy''+4y=5e-x

which is the same as the R.HS. (Right-hand side) of the given differential equation.

Hence, y=3sin2x+e-xis a solution to the differential equation y''+4y=5e-x.

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Most popular questions from this chapter

In Problems 9鈥�20, determine whether the equation is exact.

If it is, then solve it.

(2xy+3)dx+(x2-1)dy=0

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In problems 1-4Use Euler鈥檚 method to approximate the solution to the given initial value problem at the points x = 0.1, 0.2, 0.3, 0.4, and 0.5, using steps of size 0.1 (h = 0.1).

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(a) Show that y2+x-3=0 is an implicit solution todydx=-12y on the interval (-鈭�,3).

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Using the Runge鈥揔utta algorithm for systems with h = 0.05, approximate the solution to the initial value problem y'''+y''+y2=t;y(0)=1,y'(0)=0,y''(0)=1 at t=1.
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