Question

Find the general solution of the differential equation and check the result by differentiation. (Use C for the constant of integration.) dydt = 45t4y =

Answer 1

Given differential equation:

`(dy)/(dt)\text{ = 45t}^4`

Step 1: Re-arrange:

`dy\text{ = 45t}^4\text{ dt}`

Step 2: Take integral of both sides:

`\begin{gathered} \int dy\text{ = }\int45t^4dt \\ y\text{ \lparen t\rparen= 45}\int t^4dt\text{ + C} \\ y(t)\text{ = 45 }*\text{ }\frac{t^(4+1)}{4\text{ + 1}}\text{ + C} \\ y(t)\text{ = 45}*(t^5)/(5)\text{ + C} \\ y(t)\text{ = 9t}^5\text{ + C} \end{gathered}`

Check

Let us attempt to differentiate y(t):

`\begin{gathered} (dy)/(dt)\text{ = 5}*\text{ 9t}^(5-1)\text{ } \\ =\text{ 45t}^4 \end{gathered}`

Hence we have the solution of the differential equation to be:

`y(t)\text{ =9t}^5\text{ + C}`