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Category: Differential Equation

Use-Laplace-transform-to-solve-the-initial-value-problem-ty-4t-2-y-13t-4-y-0-where-y-0-0-and-y-0-0-

Question Number 116457 by bobhans last updated on 04/Oct/20 $$\mathrm{Use}\:\mathrm{Laplace}\:\mathrm{transform}\:\mathrm{to}\:\mathrm{solve}\:\mathrm{the}\: \\ $$$$\mathrm{initial}\:\mathrm{value}\:\mathrm{problem}\:\mathrm{ty}''+\left(\mathrm{4t}−\mathrm{2}\right)\mathrm{y}'+\left(\mathrm{13t}−\mathrm{4}\right)\mathrm{y}=\mathrm{0} \\ $$$$\mathrm{where}\:\mathrm{y}\left(\mathrm{0}\right)=\mathrm{0}\:\mathrm{and}\:\mathrm{y}'\left(\mathrm{0}\right)=\mathrm{0} \\ $$ Answered by Olaf last updated on 04/Oct/20 $$\mathrm{L}\left({y}''\right)\:=\:\mathrm{p}^{\mathrm{2}} \mathrm{L}\left(\mathrm{p}\right)−\mathrm{p}{y}\left(\mathrm{0}\right)−{y}'\left(\mathrm{0}\right)…

solve-the-diff-equation-1-dy-dx-x-3y-5-x-y-1-2-3y-7x-3-dx-7y-3x-7-dy-0-

Question Number 116329 by bemath last updated on 03/Oct/20 $$\mathrm{solve}\:\mathrm{the}\:\mathrm{diff}\:\mathrm{equation}\:\: \\ $$$$\left(\mathrm{1}\right)\frac{\mathrm{dy}}{\mathrm{dx}}\:=\:\frac{\mathrm{x}+\mathrm{3y}−\mathrm{5}}{\mathrm{x}−\mathrm{y}−\mathrm{1}} \\ $$$$\left(\mathrm{2}\right)\:\left(\mathrm{3y}−\mathrm{7x}−\mathrm{3}\right)\mathrm{dx}+\left(\mathrm{7y}−\mathrm{3x}−\mathrm{7}\right)\mathrm{dy}=\mathrm{0} \\ $$ Answered by mr W last updated on 03/Oct/20 $$\left(\mathrm{1}\right)\:\:\frac{{dy}}{{dx}}=\frac{{x}+\mathrm{3}{y}−\mathrm{5}}{{x}−{y}−\mathrm{1}}…

solve-the-Cauchy-Euler-Differential-Equation-by-substituting-x-e-t-x-3-d-3-y-dx-3-2x-2-d-2-y-dx-2-2y-10x-10-x-

Question Number 116105 by ShakaLaka last updated on 30/Sep/20 $${solve}\:{the}\:{Cauchy}-{Euler}\: \\ $$$${Differential}\:{Equation}\:{by} \\ $$$${substituting}\:{x}={e}^{{t}} \\ $$$${x}^{\mathrm{3}} \:\frac{{d}^{\mathrm{3}} {y}}{{dx}^{\mathrm{3}} }\:+\:\mathrm{2}{x}^{\mathrm{2}} \:\frac{{d}^{\mathrm{2}} {y}}{{dx}^{\mathrm{2}} }\:+\:\mathrm{2}{y}\:=\:\mathrm{10}{x}\:+\:\frac{\mathrm{10}}{{x}} \\ $$$$ \\…