Write down the explicit solution for each of the following: a) x’=t–sin(t); x(0)=1
b) x’+2x=4; x(0)=5
c) x’’+4x=0; x(0)=0; x’(0)=1

a) x=(t^2)/2+cos(t), b) x=2+3e^(-2t), c) x=(1/2)sin(2t)

Step-by-step explanation:

Let's solve by separating variables:

a)  x’=t–sin(t),  x(0)=1

Apply integral both sides:

where k is a constant due to integration. With x(0)=1, substitute:

Finally:

b) x’+2x=4; x(0)=5

Completing the integral:

Solving the operator:

Using algebra, it becomes explicit:

With x(0)=5, substitute:

Finally:

c) x’’+4x=0; x(0)=0; x’(0)=1

Let be the solution for the equation, then:

Substituting these equations in c)

This becomes the solution m=α±βi where α=0 and β=2

Where A and B are constants. With x(0)=0; x’(0)=1:

Finally:

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