Numerical Methods · Unit 9
Ordinary Differential Equations
Exam-focused notes for Ordinary Differential Equations (Numerical Methods, BIT203): what the TU syllabus asks and how it has actually been tested, with 9 solved past questions from this unit.
What this unit covers
- Initial value problems and solution methods
- Euler's method for ODE solving
- Heun's method for ODE solving
- Runge-Kutta fourth order method
- Boundary value problems and initial value problems
- Shooting method for boundary value problems
- Higher order differential equation solving
Runge-Kutta fourth order method
Solve $\frac{dy}{dx} = x^2 + y$, with $y(0) = 1$ for $x = 1.5$, using RK fourth order method. [5]
- ODE: $\dfrac{dy}{dx} = f(x,y) = x^2 + y$ - Initial condition: $x0 = 0$, $y0 = 1$ - Target: $y$ at $x = 1.5$ - Step size (chosen): $h = 0.5$, giving 3 steps: $0 \to 0.5 \to 1.0 \to 1.5$ $$k1 = h f(xn, yn),\quad k2 = h f\!\left(xn+\tfrac h2, yn+\tfrac{k1}{2...
Full solved answer →Boundary value problems and initial value problems
Explain boundary value problem with example. Describe how higher order differential equation can be solved. [2+3]
--- (a) Boundary Value Problem A Boundary Value Problem (BVP) is a differential equation together with a set of conditions specified at two or more different points (the boundaries of the domain), rather than all conditions given at a single point. In contr...
Full solved answer →Shooting method for boundary value problems
Solve the following ordinary differential equation using shooting method. $y'' + xy' - xy = 2x$ with boundary conditions $y(0) = 1$ and $y(2) = 10$ [10]
- ODE: $y'' + xy' - xy = 2x$ - Boundary conditions: $y(0) = 1$, $y(2) = 10$ - Interval: $[0, 2]$ - Step size (chosen for hand computation): $h = 0.5$ (4 steps) - Integration method: Euler's method --- Let $y1 = y$, $y2 = y'$. Then: $$y1' = y2$$ $$y2' = y'' ...
Full solved answer →What do you mean by boundary value problem? Use shootuing method, solve the equation: y'' = 6x², with y(0) = 1 and y(1) = 2 in the interval (0, 1) for y(0.5) taking h = 0.5[10]
- ODE: $y'' = 6x^2$ - Boundary conditions: $y(0) = 1$, $y(1) = 2$ - Interval: $(0, 1)$ - Step size: $h = 0.5$ - Required: $y(0.5)$ --- A boundary value problem (BVP) is a differential equation together with conditions specified at two or more different poin...
Full solved answer →How boundary value problems differs from initial value problems? Discuss shooting method for solving boundary value problem. [5]
Feature Initial Value Problem (IVP) Boundary Value Problem (BVP) --------- Conditions given All conditions at a single point (initial point) Conditions given at two or more different points (boundaries) Solution direction Marches forward from initial point ...
Full solved answer →Euler's method for ODE solving
Solve the following differential equation $$\frac{dy}{dx} = 3x + \frac{y}{2}$$ with $y(0) = 1$ for $x = 0.2$ $(h = 0.1)$ using Euler's Method. [5]
- ODE: $\dfrac{dy}{dx} = f(x,y) = 3x + \dfrac{y}{2}$ - Initial condition: $y(0) = 1 \Rightarrow x0 = 0,\ y0 = 1$ - Step size: $h = 0.1$ - Target: $y$ at $x = 0.2$ (2 steps) Euler's formula: $$y{n+1} = yn + h\, f(xn, yn)$$ Iteration 1: $x0 = 0,\ y0 = 1$ $$f(...
Full solved answer →Given the equation y′=3x2+1y' = 3x^2 + 1y′=3x2+1 with y(1) = 2, estimate y(2) by Euler's Method using h = 0.2. [5]
- $f(x,y) = y' = 3x^2 + 1$ - $x0 = 1.0$, $y0 = 2.0$ - $h = 0.2$ - Target: $y(2)$ → requires $\frac{2-1}{0.2} = 5$ steps Euler formula: $$y{n+1} = yn + h\,f(xn, yn)$$ Step 1 ($x0=1.0,\ y0=2.0$): $$f = 3(1.0)^2+1 = 4 \implies y1 = 2.0 + 0.2(4) = 2.800$$ Step ...
Full solved answer →Approximate the solution of y'=2x+y, y(0)=1 using Euler's method with step size of 0.1. Approximate the value of y(0.4). [5]
Given data: - ODE: $y' = f(x,y) = 2x + y$ - Initial condition: $y(0) = 1$, so $x0 = 0$, $y0 = 1$ - Step size: $h = 0.1$ - Target: $y(0.4)$ (requires 4 steps) Euler's formula: $y{n+1} = yn + h \cdot f(xn, yn)$, with $f(x,y) = 2x + y$. Step 1: $y(0.1)$ $$f(0,...
Full solved answer →Heun's method for ODE solving
Find the approximate value of y when x = 0.6 of $dy/dx = 1-2xy$, given that y = 0 when x = 0 with h=0.2 using Heun's method. [5]
- ODE: $\frac{dy}{dx} = f(x,y) = 1 - 2xy$ - Initial condition: $y = 0$ at $x = 0$ - Step size: $h = 0.2$ - Target: $y$ at $x = 0.6$ (requires 3 steps) Heun's formulas: $$y^{n+1} = yn + h\,f(xn, yn)$$ $$y{n+1} = yn + \frac{h}{2}\left[f(xn, yn) + f(x{n+1}, y^...
Full solved answer →Make Unit 9 stick
Practice BIT203 with flashcards & quizzes