Question

Find the solution r(t)r(t) of the differential equation with the given initial condition: r′(t)=⟨sin4t,sin7t,9t⟩,r(0)=⟨9,8,3⟩

Answer 1

`\mathbf r'(t)=\langle\sin4t,\sin7t,9t\rangle`

`\implies\mathbf r(t)=\displaystyle\int\mathbf r'(t)\,\mathrm dt`

`\mathbf r(t)=\displaystyle\left\langle\int\sin4t\,\mathrm dt,\int\sin7t\,\mathrm dt,\int9t\,\mathrm dt\right\rangle`

`\mathbf r(t)=\left\langle-\frac14\cos4t+C_1,-\frac17\cos7t+C_2,-\frac92t^2+C_3\right\rangle`

With
`\mathbf r(0)=\langle9,8,3\rangle`, we have


`\langle9,8,3\rangle=\left\langle-\frac14+C_1,-\frac17+C_2,C_3\right\rangle`

`\implies C_1=\frac{37}4,C_2=\frac{57}7,C_3=8`

`\implies\mathbf r(t)=\left\langle-\frac14\cos4t+\frac{37}4,-\frac17\cos7t+\frac{57}7,-\frac92t^2+8\right\rangle`