Answer to Question #243210 in Electrical Engineering for Alock kumar

Question #243210

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

1
Expert's answer
2021-09-30T02:40:10-0400

"\\mathrm{A\\:second\\:order\\:linear,\\:homogeneous\\:ODE\\:has\\:the\\:form\\:of\\:}\\:\\:ay''+by'+cy=0\\\\\n\\mathrm{For\\:an\\:equation\\:}ay''+by'+cy=0\\mathrm{,\\:assume\\:a\\:solution\\:of\\:the\\:form\\:}e^{\u03b3t}\\\\\n\\mathrm{Rewrite\\:the\\:equation\\:with\\:}y=e^{\u03b3t}\\\\\n\\left(\\left(e^{\u03b3t}\\right)\\right)''\\:-e^{\u03b3t}=0\\\\\ne^{\u03b3t}\\left(\u03b3^2-1\\right)=0\\\\\n\u03b3=1,\\:\u03b3=-1\\\\\n\\mathrm{For\\:two\\:real\\:roots\\:}\u03b3_1\\ne \\:\u03b3_2\\mathrm{,\\:the\\:general\\:solution\\:takes\\:the\\:form:\\quad }y=c_1e^{\u03b3_1\\:t}+c_2e^{\u03b3_2\\:t}\\\\\ny=c_1e^t+c_2e^{-t}\\\\\ny=e^t+e^{-t}"


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