Question

Solve the initial value problem
dx/dt−4x=cos(5t)
with x(0)=1.
x(t) = ?

Answer 1

We are given the following differential equation,
`(dx)/(dt)-4x=cos(5t)`, with the initial condition
`x(0)=1`. This is a linear DE in the form,
`(dx)/(dt)+P(t)x=Q(t)`, where
`\mu(t)=e^{\int\ {P(t)} \, dt }`.

First, finding
`\mu(t)`.

`\Longrightarrow\mu(t)=e^{\int\ {P(t)} \, dt }\Longrightarrow\mu(t)=e^{\int\ {-4} \, dt }`

`\mu(t)=e^(-4t )`

Multiply the DE by
`\mu(t)`.

`\Longrightarrow(dx)/(dt)-4x=cos(5t)\Longrightarrow e^(-4t) [(dx)/(dt)-4x=cos(5t)]`

`\Longrightarrow (e^(-4t)) (dx)/(dt)-(e^(-4t))4x=(e^(-4t))cos(5t)`

`\Longrightarrow e^(-4t) (dx)/(dt)-4e^(-4t)x=e^(-4t)cos(5t)`

Verify that the L.H.S is a product rule of
`e^(-4t)x`.

`\Longrightarrow (d)/(dt)[e^(-4t)x ] =(e^(-4t)*-4)x+(e^(-4t))((dx)/(dt) )`

`\Longrightarrow (d)/(dt)[e^(-4t)x ] =-4e^(-4t)x+e^(-4t)(dx)/(dt)`

This matches, so we can move forward by integrating both sides of the DE.

`\int\ {[e^(-4t)x ]'} \, =\int\ {e^(-4t)cos(5t)} \, dt`

The integral on the R.H.S is pretty nasty to do using integration by parts and u-sub. I will use a table of integrals.

`\int\ {e^(ax)cos(bx) } \, dx=(e^(ax)(acos(bx)+bsin(bx)))/(a^2+b^2)`

Let
`a=-4` and
`b=5`

`\int\ {[e^(-4t)x ]'} \, =\int\ {e^(-4t)cos(5t)} \, dt`

`\Longrightarrow e^(-4t)x = (e^(-4t)(-4cos(5t)+5sin(5t)))/((-4)^2+(5)^2)+C`

`\Longrightarrow e^(-4t)x = (e^(-4t)(-4cos(5t)+5sin(5t)))/(41) +C`

Now for the initial condition,
`x(0)=1`.

`\Longrightarrow e^(-4(0))(1) = (e^(-4(0))(-4cos(5(0))+5sin(5(0))))/(41) +C`

`\Longrightarrow (1)(1) = ((1)(-4(1)+(0)))/(41) +C`

`\Longrightarrow 1 = -(4)/(41) +C`

`\Longrightarrow 1+(4)/(41) = C`

`\Longrightarrow C=(45)/(41)`

Now we have,
`e^(-4t)x = (e^(-4t)(-4cos(5t)+5sin(5t)))/(41) +(45)/(41)`, solve for x.

`\Longrightarrow x = ((e^(-4t)(-4cos(5t)+5sin(5t)))/(41))/( e^(-4t)) +((45)/(41))/(e^(-4t))`

`\Longrightarrow x = ((-4cos(5t)+5sin(5t)))/(41)} +(45e^(4t))/(41)`

Thus, the answer to the given differential equation with an initial condition is,
`x = ((-4cos(5t)+5sin(5t)))/(41)} +(45e^(4t))/(41)`.