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# Numerical Solution of Ordinary Differential Equations

Consider y(t) to be a function of a variable t. A first order Ordinary differential
equation is an equation relating y, t and its first order derivatives. The most general
form is : $$F(t,y(t),y^{/} (t))=0$$
The solution with arbitrary constant is known as the general solution of the differential equation. The solution obtained using the initial condition is a particular solution.
A first order Initial Value Problem (IVP) is defined as a first order differential equation together with specified initial condition at $$t=t_0:$$
$$y=f(t,y); t_0 \leq t \leq b \text{ with } y(t_0 )= y_0$$
Sometimes it is very difficult to obtain the closed form solution of differential equation. In such cases, the approximate solution of given differential equation can be obtained.

## Euler method

Let us consider a first order & first degree differential equation as $$\frac{dy}{dx}=f(x,y)$$ with $$y(x_0 )=y_0$$. Then the Euler method to find the solution of the ODE is $$y_{n+1}=y_n+hf(x_n,y_n )=y(x_{n+1} )\forall n=0,1,2,\dots$$

Prob.(1): Given $$\frac{dy}{dx}=x^2+y^2$$ with y(0)=1. Determine y(0.3) by Euler’s Method with step length h=0.01, correct up to 4 DP.

## Exercise

1. State Euler’s formula to solve approximately $$\frac{dy}{dx}=f(x,y),y(x_0 )=y_0$$. Hence find y(0.2) with step length h=0.1 for the initial value problem $$\frac{dy}{dx}=xy ; y(0)=1$$
[C.H. 2013]
2. Given $$\frac{dy}{dx}=x^2+y$$ with y(0)=1. Determine y(0.02) by Euler’s Method with step length h=0.01, correct up to 4 significant figure.
[C.H. 2012]
3. Solve by Euler’s Method the following differential equation for x=1 by taking h=0.2, $$\frac{dy}{dx}=xy$$, y=1 when x=0. Give result correct to four decimal places.
[C.H. 2011]
4. Solve $$\frac{dy}{dx}=1-y$$ with y(0)=0, using Euler’s method. Find y at x=0.1 and x=0.2. Compare the approximate solution with exact solution.
5. Given $$\frac{dy}{dx}+\frac{y}{x}=\frac{1}{x^2}$$ with y(1)=1. Evaluate y(1.2) by Euler’s Method with step length h=0.1
6. Solve by Euler’s method the following differential equation: $$\frac{dy}{dx}=xy$$, y(0)=1 for x=1 with h=0.2
[C.H. 2017]

## Picard method

Let us consider a first order & first degree differential equation as $$\frac{dy}{dx}=f(x,y)$$ with $$y(x_0 )=y_0$$. Then the Picard’s method to find the solution of the ODE is $$y_n=y_0+ \int_{x_0}^{x} f(x,y_{n-1}) \, dx \forall n=1,2,3,\dots$$

Prob.(2): Using Picard’s method of successive approximation, obtain a solution up to fourth approximation of differential equation $$\frac{dy}{dx}=x+y$$ where y=1 when x=0
[C.H. 2011]

## Exercise

1. Solve by Picard’s method $$\frac{dy}{dx}=xy+1$$ with y(0)=1
[C.H. 2006] [C.H. 2007]
2. Solve by Picard’s method $$\frac{dy}{dx}=x-y$$ with y(0)=2
[C.H. 2009]
3. Using Picard’s method of successive approximation, obtain a solution up to fourth approximation of differential equation $$\frac{dy}{dx}=x+y$$ where y=1 when x=0
[C.H. 2011]
4. Solve by Picard’s method:$$\frac{dy}{dx}=x-y$$ with y(0)=1  and show that the sequence of approximations tends to the exact solution as a limit.
[C.H. 2008]
5. Solve $$\frac{dy}{dx}=\frac{x}{1+y^2 }$$ with the initial condition y(0)=0, by Picard’s method to obtain y(0.15); correct to two significant figures.
[C.H. 2011]
6. State Picard’s recursion formula in connection with the solution of a first order differential equation $$\frac{dy}{dx}=f(x,y)\text{ with } y_0=f(x_0)$$.
Use Picard’s method to approximate y for x=0.2 correct to four decimal places for the differential equation $$\frac{dy}{dx}=x-y$$, y(0)=1
[C.H. 2014]
7. (a) Use Picard’s method to compute y(0.1) for the IVP $$\frac{dy}{dx}=1+xy$$ given y=1 when x=0 correct to 5 significant figures.
(b) Does a single step method for solving IVP always converge? Justify your answer.
[C.H. 2017]
8. Velocity of a body is equal to the reciprocal of the displacement and time. If the displacement measured  from origin (y) at time (t)=0 initially is 1, find the displacement y(t=0.2) using Picard’s method.
[C.H. 2018]
9. Use Picard’s method to compute y for x=0.2, correct to 4 decimal places for the differential equation  $$\frac{dy}{dx}=xy+x^2$$ with end condition y(0)=0
[C.H. 2019]
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