Question

Use the Euler's method and Runge-Kutta method for systems to approximate y1(0.72+0.72), y2(0.72), y2(0.72+0.72) of the following systems of first-order differential equations:

dy1/dx=y1-y2+2, y1(0)= -1

dy2/dx= -y1+y2+4x

y2(0)=0 ; 0 ≤ x ≤ 2

Answer 1

$\mathrm{A\:second\:order\:linear,\:homogeneous\:ODE\:has\:the\:form\:of\:}\:\:ay''+by'+cy=0\\ \mathrm{For\:an\:equation\:}ay''+by'+cy=0\mathrm{,\:assume\:a\:solution\:of\:the\:form\:}e^{γt}\\ \mathrm{Rewrite\:the\:equation\:with\:}y=e^{γt}\\ \left(\left(e^{γt}\right)\right)''\:-e^{γt}=0\\ e^{γt}\left(γ^2-1\right)=0\\ γ=1,\:γ=-1\\ \mathrm{For\:two\:real\:roots\:}γ_1\ne \:γ_2\mathrm{,\:the\:general\:solution\:takes\:the\:form:\quad }y=c_1e^{γ_1\:t}+c_2e^{γ_2\:t}\\ y=c_1e^t+c_2e^{-t}\\ y=e^t+e^{-t}$