00:01
In this question, first we need to find the derivative of y with respect to t.
00:05
So which is given by what? which is given by y dash is equals to z dash plus we have t multiply by z double dash.
00:17
Next we substitute y and y dash into the differential equation.
00:23
So here t ez dash is equals to y is equals to z dash plus we have what? we have t multiply by z double dash.
00:38
So simplifying this equation we get what? so we get z double dash plus we have 1 divided by t multiply by z dash.
00:52
This is equals to e to the power minus t.
00:55
So this is the second order linear differential equation with variable coefficient.
01:00
So to solve it we can use the method of undetermined coefficient.
01:05
So we assume a particular solution from zp.
01:11
So here z rho is equals to a multiply by e to the power minus t.
01:20
So substituting this into the differential equation we get what? so we get a multiply by e to the power minus t minus we have a multiply by e to the power minus t whole divided by t.
01:45
This is equals to what? this is equals to e to the power t e to the power minus t.
01:51
So solving for a now solving for a we get what? so we get a is equals to 1 divided by 1 divided by t minus 1.
02:08
So this is equals to what? this is equals to t...