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

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

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

VeritasVosLiberabit
 9 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
 9 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
 9 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
 9 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
 9 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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