Solve the initial value problem: $$ \frac{d y}{d x}=7 y, \quad y(0)=4 $$ $$ y(x)= $$

Question

UpStudy AI · Accepted Answer

1. Separate variables:

$$
   \frac{1}{y} \, dy = 7 \, dx
   $$

2. Integrate both sides:

$$
   \int \frac{1}{y} \, dy = \int 7 \, dx
   $$
   
   $$
   \ln|y| = 7x + C
   $$

3. Exponentiate to solve for $$ y $$:

$$
   |y| = e^{7x+C} = e^{7x} \cdot e^C
   $$
   
   Let $$ C_1 = e^C $$, then

$$
   y = C_1 e^{7x}
   $$

4. Apply the initial condition $$ y(0)=4 $$:

$$
   4 = C_1 e^{7\cdot 0} = C_1 \quad \Rightarrow \quad C_1 = 4
   $$

Thus, the solution is:

$$
y(x)= 4 e^{7x}
$$
$$ y(x) = 4 e^{7x} $$

Solve the initial value problem: