199k views
4 votes
Not sure how I am supposed to arrive to an answer

Not sure how I am supposed to arrive to an answer-example-1
User Vishu
by
4.2k points

1 Answer

1 vote

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}

User Numan Tariq
by
4.5k points