Question

If $\frac{dy}{dx}=y+3$ and $y(0)=2$, then $y(\log 2)$ is equal to

• A. -2
• B. 2
• C. 5
• D. 7

Answer 1

Answer: (D)

7

Answer 2

The given differential equation is $\frac{dy}{dx} = y+3$. This is a first-order differential equation that can be solved using separation of variables.

1. Separate the variables:

   $\frac{dy}{y+3} = dx$

2. Integrate both sides:

   $\int \frac{dy}{y+3} = \int dx$

   $\log|y+3| = x + C$

   where C is the constant of integration.

3. Use the initial condition to find the constant C:

   We are given $y(0)=2$. Substitute $x=0$ and $y=2$:

   $\log|2+3| = 0 + C$

   $\log 5 = C$

4. Substitute the value of C back into the general solution:

   $\log|y+3| = x + \log 5$

5. Express y explicitly:

   $\log|y+3| - \log 5 = x$

   Using the logarithm property $\log a - \log b = \log(a/b)$:

   $\log\left|\frac{y+3}{5}\right| = x$

   Exponentiate both sides with base $e$:

   $\frac{y+3}{5} = e^x$

   (Since $y(0)=2$, $y+3=5$, which is positive. Assuming a continuous solution, $y+3$ remains positive, so $|y+3|=y+3$.)

   $y+3 = 5e^x$

   $y = 5e^x - 3$

6. Calculate $y(\log 2)$:

   Substitute $x = \log 2$:

   $y(\log 2) = 5e^{\log 2} - 3$

   Using the property $e^{\log a} = a$:

   $y(\log 2) = 5(2) - 3$

   $y(\log 2) = 10 - 3$

   $y(\log 2) = 7$