Given $y(t)$ is a solution of $(1+t)\large\frac{dy}{dt}$$-ty=1$ and $y(0)=-1$
Consider the equation
$(1+t) \large\frac{dy}{dt}$$-ty=1$
This can be written as
$\large\frac{dy}{dt}-\frac{t}{1+t}$$.y=\large\frac{1}{1+t}$
By dividing throughout by $1+t$
Clearly this is a linear differential equation of the form
$\large\frac{dy}{dx}$$+Py=Q$
Here $P=\large\frac{-t}{1+t}$ and $Q=\large\frac{1}{1+t}$
Let us first the integrating factor I.F
$e^{\large\int pdx}=e^{\int \Large\frac{-t}{1+t}dt}$
Consider $ I=\int \large\frac{-t}{1+t} $$dt$
Add +1 and subtract 1 to the numerator
$ -\int \large \frac{t+1-1}{1+t}dt$
On spliting the terms we get,
$-\bigg[\int dt-\int \large\frac{1}{1+t}$$dt\bigg]$
Step 2:
On integrating we get,
$-[t-\log(1+t)]$
Now I.F is
$e ^{\large -t+\log(1+t)}=e^{-t}.e^{\log (1+t)}$
But $ e^{\large\log x}$$=x$
Therefore $I.F=e^{-t}(1+t)$
Hence the required solution is
$y e^{\int pdx}=Q . e^{\int pdx}dt +c$
=>$ye^{-t}.(1+t)=\int \large\frac{1}{1+t}.$$e^{-t}(1+t)dt+c$
$ye^{-t}(1+t)=\int e^{-t}.dt$
Step 3:
On integrating we get,
$ye^{-t}(1+t)=-e^{-t}+c$
Dividing throughout by $e^{-t}$ we get
$y(1+t)=-1+ce^{-t}$
It is given $y(0)=-1$
=>When $t=0,y=-1$
Let us substitute the above values of x and y to obtain the value of c
$(-1)(1+0)=-1+ce^0$
But $e^0=1$
We are asked to prove that the value when
$y(1)=-1/2$
Hence $-1=-1+c=>c=0$
Hence the equation is $y(1+t)=-1$
$y(1)=>t=1$
When $t=1$ we know $c=0$
$y(1+1)=-1$
=>$2y=-1$ Therefore $y=\large\frac{-1}{2}$
Hence $y(1)=\large\frac{-1}{2}$ is proved.