Numerical solution of first order ordinary differential equations

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Numerical solution of first order ordinary differential equations

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]