Question

Not sure how I am supposed to arrive to an answer

Answer 1

Integrating both sides once gives

`(\mathrm d\mathbf r)/(\mathrm dt)=2e^t\,\mathbf i+3e^(-t)\,\mathbf j+4e^(2t)\,\mathbf k+\mathbf c`

where
`\mathbf c` is an arbitrary constant vector. Use the initial condition to find its value:

`(\mathrm d\mathbf r)/(\mathrm dt)(0)=-\mathbf i+7\,\mathbf j=(2+c_1)\,\mathbf i+(3+c_2)\,\mathbf j+(4+c_3)\,\mathbf k`

`\implies\mathbf c_1=-3\,\mathbf i+4\,\mathbf j-4\,\mathbf k`

Integrate again:

`\mathbf r(t)=2e^t\,\mathbf i-3e^(-t)\,\mathbf j+2e^(2t)\,\mathbf k+\mathbf c_1t+\mathbf c_2`

where
`\mathbf c_2` is another arbitrary vector of constants. Use the other initial condition to determine its components:

`\mathbf r(0)=6\,\mathbf i+\mathbf j+3\,\mathbf k=(2+c_1)\,\mathbf i+(-3+c_2)\,\mathbf j+(2+c_3)\,\mathbf k`

`\implies\mathbf c_2=4\,\mathbf i+4\,\mathbf j+\mathbf k`

Then the particular solution to this ODE is

`\boxed{\mathbf r(t)=(2e^t-3t+4)\,\mathbf i+(-3e^(-t)+4t+4)\,\mathbf j+(2e^(2t)-4t+1)\,\mathbf k}`