**dv/dt = p(t) v + q(t)**Where,

**p(t) = 5(1+t)***and,*

**q(t) = (1+t)e-t**
The initial value is,

*v(0) = 1*; and the time period is**.***0 < t < 10*
Implicit Euler approach is unconditionally stable. The implementation of Implicit Euler scheme may be represented as,

The following plot shows the progression of the solution as the time increases.

*v_n+1 = (v_n + hq_n+1) / (1 + hp_n+1)*
The following MATLAB program implements this scheme:

**MATLAB program:**

**close all;**

**clc;**

**y1(1) = 0; % Initial condition**

**h = 0.1; % Time step**

**t1(1) = 0;**

**i = 1;**

**% Implementation of the Implicit Euler approach**

**while (t1(i) <= 30)**

**q=(1+t1(i))*exp(-t1(i));**

**p= 5*(1+t1(i));**

**y1(i+1) = (y1(i)+(h*q))/(1+h*p);**

**t1(i+1) = t1(i) + h;**

**i = i + 1;**

**end**

**plot(t1,y1,'-o')**

**hold on**

**Program Output:**

The following plot shows the progression of the solution as the time increases.