1 a Verify that y is a solution of the ODE b Determine from

1. (a) Verify that y is a solution of the ODE. (b) Determine from y the particular solution of the IVP.

Solution

a)

y = ce^-4x+0.35

y\' = -4ce^-4x = -4(y-0.35)

= -4y+1.4

As y satisfies the DE , it is verified that y is a solution of the ODE>

To find c

y(0) =2

i.e. 2 = ce^-4(0)+0.35

1.65 = c

Hence particular solution is

y = 1.65e^-4x+0.35

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b) y\' = y + e^x

y = (x+c)e^x

= xe^x+ce^x

y\' = e^x+xe^x+ce^x

= e^x+y

Thus verified that y is a solution of the ODE.

To find c:

y(0) = 0.5. Plug in these values

0.5 = ce⁰

c= 0.5

Hence solution is

y = (x+0.5) e^x