Question Type 1: Euler’s approximation method for differential equations Exercises

Use Euler's method with step size $h=-0.1$ to approximate the solution to $\frac{dy}{dx}=y-x$ with $y(0)=1$ at $x=-0.1$.

Use Euler's method with step size $h=0.2$ to approximate the solution to $\frac{dy}{dx}=x+y$ with $y(0)=2$ at $x=0.4$.

Use Euler's method with $h=0.1$ to approximate the solution to $\frac{dy}{dx}=x^2+y^2$ with $y(0)=1$ at $x=0.2$.

Use Euler's method with $h=0.1$ to approximate $y(0.3)$ for the differential equation $\frac{dy}{dx}=x\,e^y$ with initial condition $y(0)=0$.

Use Euler's method with $h=0.05$ to approximate the solution to $\frac{dy}{dx}=x\,y$ with $y(0)=1$ at $x=0.1$.

Use Euler's method with $h=0.1$ to approximate the solution to $\frac{dy}{dx}=\sin(x)+y$ with $y(0)=0$ at $x=0.2$.

Use Euler's method with step size $h=0.1$ to approximate the solution to the differential equation $\frac{dy}{dx}=y$ with initial condition $y(0)=1$ at $x=0.1$.

Use Euler's method with step size $h=0.05$ to approximate the solution to $\frac{dy}{dx}=x^2+y^2$ with $y(0)=1$ at $x=0.4$.

Use Euler's method with $h=0.1$ to approximate the solution to $\frac{dy}{dx}=x^2+y^2$ with $y(0)=1$ at $x=0.4$.

Use Euler's method with step size $h=0.5$ to approximate $y(2)$ for the differential equation $\frac{dy}{dx}=x^2-y$ with initial condition $y(1)=0$.

Use Euler's method with $h=0.2$ to approximate $y(0.4)$ for the differential equation $\frac{dy}{dx}=e^{-x}-2y$ with $y(0)=1$.

Use Euler's method with $h=0.1$ to approximate $y(0.2)$ for the differential equation $\frac{dy}{dx}=y$ with $y(0)=1$, then compute the absolute error compared to the exact solution.