Consider the initial value problem
$dy/dx=x+y^2$
with $y(0)=1$
a) Use Euler's Method with step-length $h=0.1$ to find an approximation to $y(0.3)$.
HINT 1: :Numerical methods.
HINT 2: Differential equations videos.
b) Let $P2(x)$ denote the second order Taylor polynomial for the solution of the initial value problem $y(x)$ at $x=0$. Find $P2(0.3)$. HINT: Differentiate the differential equation implicitly to find $y′′$.
I just want to know if I've done it correctly.
$y′(0)=0+1^2=1$
$y′′(0)=1+2⋅1⋅1=3$
y(0.3) by Euler method:
$y1=y0+hf(x0,y0)=1+0.1(0⋅1^2)=1$
$y2=y1+hf(x1,y1)=1+0.1(0.1⋅1^2)=1.01$
$y3=y2+hf(x2,y2)=1.01+0.1(0.2⋅1.01^2)=1.030402$
Inserting into Taylor formula:
$P2(0.3)=1+1\cdot (0.3-0)+\frac{3\cdot (0.3-0)^2}{2!}=1.435$
Is this correct? Shouldn't the result from $P2(0.3)$ be closer to that of the Euler method?
Best Answer
Given:
$$\tag 1 \dfrac{dy}{dx}=x+y^2, y(0) = 1, h = 0.1$$
For $(1)$, using Euler's Method we have:
Thus, the iterates are:
Next, you need to read what is being asked for in the Taylor Polynomial approach and rework that. This is a different approach than the Euler approach.
They provide a hint for this, implicitly differentiate the DEQ to find $y''$. Implicitly differentiating $(1)$ yields:
$$y'' = 1 + 2 y y' = 1 + 2 y (x+y^2) = 1 + 2 x y + 2 y^3$$
Hopefully, you can take it from here.