Given differential equation:
[tex]\frac{dy}{dt}\text{ = 45t}^4[/tex]
Step 1: Re-arrange:
[tex]dy\text{ = 45t}^4\text{ dt}[/tex]
Step 2: Take integral of both sides:
[tex]\begin{gathered} \int dy\text{ = }\int45t^4dt \\ y\text{ \lparen t\rparen= 45}\int t^4dt\text{ + C} \\ y(t)\text{ = 45 }\times\text{ }\frac{t^{4+1}}{4\text{ + 1}}\text{ + C} \\ y(t)\text{ = 45}\times\frac{t^5}{5}\text{ + C} \\ y(t)\text{ = 9t}^5\text{ + C} \end{gathered}[/tex]
Check
Let us attempt to differentiate y(t):
[tex]\begin{gathered} \frac{dy}{dt}\text{ = 5}\times\text{ 9t}^{5-1}\text{ } \\ =\text{ 45t}^4 \end{gathered}[/tex]
Hence we have the solution of the differential equation to be:
[tex]y(t)\text{ =9t}^5\text{ + C}[/tex]
