GATE (TF) Textile 2021 Question Paper Solution | GATE/2021/TF/15

Question 15 (Textile Technology & Fibre Science)

If the numerical solution of the initial value problem
$y' = \frac{t^2}{t+y^3}$ , $y(0) = 1$ ,
is obtained by Euler’s method with step size of 0.2, then the value of y(0.4), (rounded off to two decimal places), is ___0.98 to 1.10___.

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Given-

$y' = \frac{t^2}{t+y^3}$ y(0)=1 , h=0.2 ,y(0.4)=?

$f(t,y)=\frac{t^2}{t+y^3}$

Eulers method yields-

$y_1=y_0+hf(x_0 , y_0)$ x=0 , y(0)=1

When h=0.2 $y_1=1+0.2f(0 , 1)$ =1+0.2(0)=1

When h=0.4 $y_2=y_1+hf(0.2 , 1) =1+0.4(\frac{0.2^2}{0.2+1})$ =1.006 (Ans)

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What is Euler’s method ?

Euler’s method is a numerical method for solving ordinary differential equations (ODEs). ODEs describe the behavior of systems that change over time, such as the motion of a falling object or the growth of a population.
Euler’s method approximates the solution to an ODE by computing the value of the function at discrete time intervals, or steps, and using this information to predict the value of the function at the next time step. The method is based on the idea that the derivative of a function at a point represents the slope of the tangent line to the function at that point. Using this idea, Euler’s method approximates the slope of the tangent line at a given point and uses this slope to extrapolate the value of the function at the next time step.
Mathematically, Euler’s method can be described by the following formula:
y[i+1] = y[i] + h*f(x[i], y[i])
where y[i] is the approximation of the function at the ith time step, h is the step size, f(x[i], y[i]) is the derivative of the function at the ith time step, and y[i+1] is the prediction of the function at the (i+1)th time step.
Euler’s method is a simple and easy-to-implement method for solving ODEs, but it has some limitations. In particular, the accuracy of the method depends on the step size, with smaller step sizes generally producing more accurate results. However, using smaller step sizes can increase the computational cost of the method, making it less efficient for large or complex problems. There are also more sophisticated numerical methods, such as Runge-Kutta methods, that can provide higher accuracy and better stability for certain types of ODEs.

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