Question #224866

Solve d³y/dx³-2 d²y/dx²+ dy/dx=sinh.x+2019

Expert's answer

Characteristic equation given by

"x^3 - 2 x^2 +x - \\sinh x-2019= 0"

Th general solution

"y'''\\:-2y''\\:+y=0\\\\\n\\mathrm{Rewrite\\:the\\:equation\\:with\\:}y=e^{\u03b3t}\\\\\n\\left(\\left(e^{\u03b3t}\\right)\\right)'''\\:-2\\left(\\left(e^{\u03b3t}\\right)\\right)''\\:+e^{\u03b3t}=0\\\\\ne^{\u03b3t}\\left(\u03b3^3-2\u03b3^2+1\\right)=0\\\\\n\u03b3=1,\\:\u03b3=\\frac{1+\\sqrt{5}}{2},\\:\u03b3=\\frac{1-\\sqrt{5}}{2}\\\\\n\\mathrm{For\\:non\\:repeated\\:real\\:roots\\:}\u03b3_1,\\:\u03b3_2,\\:...,\\:\u03b3_n\\mathrm{,\\:the\\:general\\:solution\\:takes\\:the\\:form:}\\\\\ny=c_1e^{\u03b3_1\\:t}+c_2e^{\u03b3_2\\:t}+...+c_ne^{\u03b3_n\\:t}\\\\\nc_1e^t+c_2e^{\\frac{1+\\sqrt{5}}{2}t}+c_3e^{\\frac{1-\\sqrt{5}}{2}t}\\\\\ny=c_1e^t+c_2e^{\\frac{\\left(1+\\sqrt{5}\\right)t}{2}}+c_3e^{\\frac{\\left(1-\\sqrt{5}\\right)t}{2}}\\\\"

Particular solution

"y_p=\\frac{\\sinh t +2019}{D^3-2D^2+D}\\\\\n=\\frac{1}{2}[1+t-\\frac{1}{6}(\\sinh t)]\\\\\n\\therefore y= c_1e^t+c_2e^{\\frac{\\left(1+\\sqrt{5}\\right)t}{2}}+c_3e^{\\frac{\\left(1-\\sqrt{5}\\right)t}{2}}+\\frac{1}{2}[1+t-\\frac{1}{6}(\\sinh t)]\\\\"

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