Problem 5 - Active Calculus

Question

Find an equation of the curve that satisfies $$ \frac{d y}{d x}=45 y x^{4} $$ and whose \(y\) -intercept is \(5 .\) \(y(x)=\)

Step-by-Step Solution

Answer

The equation of the curve is \(y = 5 e^{9x^5}\).

Step 1: Write Down the Differential Equation

The given differential equation is: \(\frac{dy}{dx} = 45yx^4\).

Step 2: Separate the Variables

To solve the differential equation, separate the variables by dividing both sides by \(y\) and multiplying both sides by \(dx\): \(\frac{1}{y} dy = 45x^4 dx\).

Step 3: Integrate Both Sides

Integrate both sides to find the general solution: \(\int\frac{1}{y} dy = \int 45x^4 dx\). This gives: \(\text{ln}|y| = \frac{45}{5}x^5 + C\). Simplifying, we get: \(\text{ln}|y| = 9x^5 + C\).

Step 4: Exponentiate to Remove the Natural Log

Exponentiate both sides to solve for \(y\): \(e^{\text{ln}|y|} = e^{9x^5 + C}\). This simplifies to \(|y| = e^{9x^5} e^C\). Let \(e^C = C'\), then \(y = C' e^{9x^5}\).

Step 5: Determine the Constant Using the Initial Condition

Use the given initial condition \(y(0) = 5\) to find the constant \(C'\). Substituting \(x = 0\) and \(y = 5\) into the equation: \(5 = C' e^{0}\). Therefore, \(C' = 5\).

Step 6: Write the Final Solution

Substitute \(C' = 5\) back into the equation: \(y = 5 e^{9x^5}\).

Key Concepts

Separation of Variables

To solve the differential equation \(\frac{dy}{dx} = 45yx^4\), we use a method called separation of variables. This technique allows us to separate the terms involving \(y\) and \(x\) on opposite sides of the equation.
First, we rewrite the equation to isolate \(dy\) and \(dx\). By dividing both sides by \(y\) and multiplying both sides by \(dx\), we get:
\( \frac{1}{y} dy = 45x^4 dx \).
Now, the terms involving \(y\) are on one side and those involving \(x\) are on the other, making it possible to integrate each side independently.

Integration

The next step after separating the variables is integration. We integrate both sides of the separated equation to find the general solution.
Integrating \(\frac{1}{y} dy\) with respect to \(y\) and \(45x^4 dx\) with respect to \(x\), we get:
\(\text{ln}|y| = \frac{45}{5}x^5 + C\).
We simplify this to:
\(\text{ln}|y| = 9x^5 + C\)
where \(C\) is the constant of integration.
To solve for \(y\), we exponentiate both sides to remove the natural logarithm, leading to:
\(|y| = e^{9x^5 + C}\).
By letting \(e^C = C'\), a new constant, we find:
\(y = C' e^{9x^5}\).

Initial Conditions

Initial conditions help determine the specific solution of a differential equation. In our example, the initial condition given is \(y(0) = 5\).
We use this to find the constant \(C'\) in our general solution \(y = C' e^{9x^5}\).
Substituting \(x = 0\) and \(y = 5\), we get:
\(5 = C' e^{0}\).
This simplifies to \(C' = 5\).
Finally, substituting back into the general solution, we obtain the specific solution:
\(y = 5 e^{9x^5}\).
Initial conditions are a crucial part of solving differential equations as they enable us to find a unique solution that fits the given context.