Question

Write and solve the differential equation that models the verbal statement. Evaluate the solution at the specified value of the independent variable. The rate of change of \(y\) is proportional to \(y .\) When \(x=0, y=4\) and when \(x=3, y=10 .\) What is the value of \(y\) when \(x=6 ?\)

Short answer

The value of y when x=6 is 25.

Step-by-step solution

Write Differential Equation

Knowing the underlying form of this problem, one can write the differential equation as \(\frac{dy}{dx} = k * y\). Remember, k is the constant of proportionality.

Solve the Differential Equation

To solve this differential equation, one can separate variables and integrate. The result is \(ln|y| = kx + C\), where C is the constant of integration. To get y by itself, one should exponentiate both sides giving \(y = e^{kx+C}\). Since \(e^{kx+C}\) equals \(e^{kx} * e^{C}\), the equation can rewrite as \(y = A * e^{kx}\), where \(A = e^C\) is a new constant.

Find Constant A Using Initial Condition

The problem gives the initial condition that when x=0, y=4. Substituting these values into the solution derived above, one gets \(4 = A * e^{k*0} = A\). Hence, \(A = 4\), and so the differential equation has the particular solution \(y = 4 * e^{kx}\).

Find Constant k Using Another Condition

It is also known that x=3 when y=10. By substituting these values into the solution equation \(y = 4 * e^{3k} = 10\), one gets \(k = ln(2.5)/3\). The particular solution can be rewritten as \(y = 4 * e^{ln(2.5)x/3}\).

Find y When x=6

Finally, to find the value of y when x=6, plug \(x = 6\) into the final solution to get \(y = 4 * e^{ln(2.5)*6/3} = 4*2.5^2 = 4*6.25 = 25\).