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qila
 11 months ago
verify that y=(4e^3x) 2 is an explicit solution of differential equation y'3y=6
qila
 11 months ago
verify that y=(4e^3x) 2 is an explicit solution of differential equation y'3y=6

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esamalaa
 11 months ago
Best ResponseYou've already chosen the best response.0use the variable separation to solve this D E

qila
 11 months ago
Best ResponseYou've already chosen the best response.0can you try to solve it and explain more about it?

esamalaa
 11 months ago
Best ResponseYou've already chosen the best response.0i think that's the right solution

VeritasVosLiberabit
 11 months ago
Best ResponseYou've already chosen the best response.0\[\frac{ dy }{ dx }=6+3y\] \[\frac{ dy }{ 6+3y }=dx\] \[\int\limits_{}^{}\frac{ dy }{ 6+3y }=\int\limits_{}^{}dx\] \[\frac{ 1 }{ 3 }\ln(6+3y)+C _{1}=x+C _{2}\]

VeritasVosLiberabit
 11 months ago
Best ResponseYou've already chosen the best response.0\[\ln(6+3y)=3x+C\] \[e ^{\ln(6+3y)}=e ^{3x+C}\] \[6+3y=e ^{3x}e ^{C}\] \[3y=Ce ^{3x}6\] \[y=Ce ^{3x}2\]

VeritasVosLiberabit
 11 months ago
Best ResponseYou've already chosen the best response.0I think you need to know an intial value y_0 to find C = 4.

UnkleRhaukus
 11 months ago
Best ResponseYou've already chosen the best response.1This question does not ask you to solve the DE, it asks to verify that a given solution solves the DE.

UnkleRhaukus
 11 months ago
Best ResponseYou've already chosen the best response.1The function \[y(x)=4e^{3x} 2\] its derivative\[y'(x) = 12e^{3x}\] The differential equation \[\qquad y'3y\qquad \quad =6\] Plugging the function and its derivative into the DE\[\begin{align} [12e^{3x}]3[4e^{3x} 2]&=6\\ 12e^{3x}12e^{3x} +6&=6\\ 6&=6\\ 0&=0 \end{align}\](a true statement) Hence \(y=4e^{3x} 2\) is an explicit solution to the differential equation.
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