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

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1. esamalaa

use the variable separation to solve this D E

2. qila

can you try to solve it and explain more about it?

3. esamalaa

i think that's the right solution

4. VeritasVosLiberabit

$\frac{ dy }{ dx }=6+3y$ $\frac{ dy }{ 6+3y }=dx$ $\int\limits_{}^{}\frac{ dy }{ 6+3y }=\int\limits_{}^{}dx$ $\frac{ 1 }{ 3 }\ln(6+3y)+C _{1}=x+C _{2}$

5. VeritasVosLiberabit

$\ln(6+3y)=3x+C$ $e ^{\ln(6+3y)}=e ^{3x+C}$ $6+3y=e ^{3x}e ^{C}$ $3y=Ce ^{3x}-6$ $y=Ce ^{3x}-2$

6. VeritasVosLiberabit

I think you need to know an intial value y_0 to find C = 4.

7. UnkleRhaukus

This question does not ask you to solve the DE, it asks to verify that a given solution solves the DE.

8. UnkleRhaukus

The 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.