# Consider that math modeling following initial valu problem dy/dt=3-2t-0.5y, y(0)=1

Consider that math modeling following initial valu problem
$\frac{dy}{dt}=3-2t-0.5y,y\left(0\right)=1$
We would liketo find an approximation solution with the step size $h=0.05$
What is the approximation of $y\left(0.1\right)?$

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Given Initial Value Problem is:
$\frac{dy}{dt}=3-2t-0,5y,y\left(0\right)=1,{t}_{0}=0$
and step size $h=0,05$
Here $f\left(t,y\right)=3-2t-0,5y$
$f\left({t}_{n-1},{y}_{n-1}\right)=3-2{t}_{n-1}-0,5{y}_{n-1}$
$=3-2\left\{{t}_{0}+\left(n-1\right)h\right\}-0,5{y}_{n-1}$
Now we have ${t}_{0}=0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{y}_{0}=1$
for $n=1$ $\therefore f\left({t}_{0},{y}_{0}\right)=3-2\left\{0+\left(1-1\right)h\right\}-0,5$
$=3-2×0-0,5×1$
$=3-0,5$
$=2,5$
$\therefore {y}_{1}={y}_{0}+hf\left({t}_{0},{y}_{0}\right)$
$=1+0,05×2,5$
$=1+0,125$
$=1,125$
Now, ${t}_{1}={t}_{0}+\left(n-1\right)h=0+\left(2-1\right)×0,05=0,05,{f}_{0|><}n=2{f}_{0|><}$
$n=2,$
$f\left({t}_{1},{y}_{1}\right)=3-2\left\{{t}_{0}+\left(n-1\right)h\right\}-0,5{y}_{1}$
$=3-2\left\{0+\left(2-1\right)0,05\right\}-0,5×1,125$
$=3-2×0,05-0,5625$
$=3-0,1-0,05625$
$=2,3375$
$\therefore {y}_{2}={y}_{1}=hf\left({t}_{1},{y}_{1}\right)$
$=1,125+0,05×2,3375$