dy/dx = 3y and y(0) = 3. y(ln 2) =

Answer

Explanation:

Step1: Separate variables

Separate the variables in the differential - equation $\frac{dy}{dx}=3y$. We get $\frac{dy}{y}=3dx$.

Step2: Integrate both sides

Integrate $\int\frac{dy}{y}=\int3dx$. The left - hand side integral is $\ln|y|$ and the right - hand side is $3x + C$. So, $\ln|y| = 3x+C$.

Step3: Use the initial condition

Given $y(0) = 3$. Substitute $x = 0$ and $y = 3$ into $\ln|y|=3x + C$. We have $\ln(3)=3\times0 + C$, so $C=\ln(3)$.

Step4: Find the function $y(x)$

The equation becomes $\ln|y|=3x+\ln(3)$. Exponentiating both sides gives $y = 3e^{3x}$ (since $y>0$ as $y(0)=3$).

Step5: Evaluate $y(\ln 2)$

Substitute $x=\ln 2$ into $y = 3e^{3x}$. Then $y(\ln 2)=3e^{3\ln 2}$. Since $3\ln 2=\ln(2^{3})=\ln 8$, and $e^{\ln 8}=8$, we have $y(\ln 2)=3\times8 = 24$.