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:

$\frac{dy}{dt}\text{ = 45t}^4$

Step 1: Re-arrange:

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

Step 2: Take integral of both sides:

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

Check

Let us attempt to differentiate y(t):

$\begin{gathered} \frac{dy}{dt}\text{ = 5}\times\text{ 9t}^{5-1}\text{ } \\ =\text{ 45t}^4 \end{gathered}$

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

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