Math Challenges

Submissions for Problem #36

Use Euler's method to approximate four points (beyond y0) of the solution in the interval [0,1] for the given differential equation below.

\[ \displaylines{\frac{dy}{dt}=t^2+y\\ y\left(0\right)=1\\ \Delta t=0.25} \]

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Solution: \( \displaylines{y_{n+1}=y_{n}+f\left(t_{n},y_{n}\right)\Delta t\\ \\ f\left(t,y\right)=t^2+y,\Delta t=0.25\\ y_0=1\text{ and }t_0=0,t_1=0.25,\text{etc.}\\ \\ y_1=y_0+\left\lbrack t_0^2+y_0\right\rbrack\Delta t=1+\left\lbrack1\right\rbrack\left(0.25\right)=1.25\\ y_2=1.25+\left\lbrack1.3125\right\rbrack\left(0.25\right)\approx1.578\\ y_3=1.578+\left\lbrack1.828\right\rbrack\left(0.25\right)\approx2.035\\ y_4=2.035+\left\lbrack2.598\right\rbrack\left(0.25\right)\approx2.685} \)