Question

The solution of the differential equation satisfying initial condition \(y(0)=1\) is given. \(y^{\prime}=-4 t^{3} y ; \quad y(t)=e^{-t^{4}}\)

Short answer

Question: Verify that the given solution \(y(t) = e^{-t^4}\) satisfies the differential equation \(y^{\prime}=-4 t^{3} y\) and the initial condition \(y(0)=1\). Short Answer: After finding the derivative of the given solution \(y^{\prime}(t) = -4t^3 e^{-t^4}\) and substituting it into the differential equation, we observe that the given solution satisfies the equation. Furthermore, the initial condition \(y(0)=1\) is also satisfied. Therefore, \(y(t) = e^{-t^4}\) is a valid solution to the given differential equation and initial condition.

Step-by-step solution

Find the derivative of the given solution

To find the derivative of the given solution, we use the chain rule. The given solution is \(y(t) = e^{-t^4}\), so we have: \(y^{\prime}(t) = \dfrac{d}{dt}(e^{-t^4}) = -4t^3 e^{-t^4}\)

Verify that the differential equation is satisfied

Now we need to check if the obtained derivative, \(y^{\prime}(t)=-4t^3 e^{-t^4}\), satisfies the given differential equation \(y^{\prime}=-4 t^{3} y\). So let's substitute \(y(t)\) into the equation and see if both sides are equal: Left side: \(y^{\prime} = -4t^3 e^{-t^4}\) (from Step 1) Right side: \(-4t^3 y = -4t^3 (e^{-t^4})\) As we can see, both sides of the equation are equal, so the given solution satisfies the differential equation.

Check if the initial condition is satisfied

Finally, we need to verify if the given solution also satisfies the initial condition \(y(0)=1\). To do this, we will substitute \(t=0\) in the given solution and see if it equals to 1: \(y(0) = e^{-0^4} = e^0= 1\) The given solution satisfies the initial condition \(y(0)=1\). So, we have successfully verified that the solution \(y(t) = e^{-t^4}\) satisfies both the given differential equation (\(y^{\prime}=-4 t^{3} y\)) and the initial condition (\(y(0)=1\)).

Key concepts

Chain Rule

When it comes to solving differential equations, one indispensable mathematical tool is the chain rule. This rule enables us to differentiate composite functions—functions that are nested within each other, like a Russian doll. For example, if we have a function h = g(f(t)), where g is a function of f, and f is a function of t, then using the chain rule, the derivative of h with respect to t is the product of the derivative of g with respect to f and the derivative of f with respect to t, denoted as \( h'(t) = g'(f(t)) \cdot f'(t) \).

Let's make it simpler: imagine you're following directions to get to a treasure. You can only proceed to the next step after completing the one before it—this 'step-by-step' approach is similar to how the chain rule works. Breaking down the process like this can make it much more manageable, just as our step-by-step solution illustrated. By using the chain rule, we differentiated the given solution \( y(t) = e^{-t^4} \) with respect to t, yielding \( y'(t) = -4t^3 \cdot e^{-t^4} \). This is what allowed us to check if the solution satisfies the differential equation in the subsequent steps.

Initial Condition

The concept of an initial condition is akin to knowing the starting point on a map before embarking on a journey. In the realm of differential equations, the initial condition is a point on the solution curve that is known to us, typically expressed as \( y(a) = b \), where a is the initial time and b is the initial value.

For example, in the exercise we have the initial condition \( y(0) = 1 \), stating that at time \( t=0 \) the value of our solution \( y \) is 1. Verifying this condition is crucial because it ensures that the solution is not just an abstract formula but rather a specific function that passes through a given point. This aspect is like having a key to the treasure chest—you must ensure it's the correct one to unlock the treasure! By substituting \( t=0 \) into our solution \( y(t) \), we confirm that the treasure chest unlocks, meaning our solution fulfills the given initial condition and is therefore the correct key.

Exponential Function

An exponential function is a powerful mathematical expression, often denoted by \( y = e^x \), where \( e \) is an irrational and transcendental number approximately equal to 2.71828. The beauty of exponential functions lies in their consistency: the rate of growth (or decay) is proportional to their current value. This property makes them extremely useful in modeling phenomena that grow or decrease rapidly and continuously, such as populations, investments, or radioactive decay.

In our exercise, the given solution \( y(t) = e^{-t^4} \) involves an exponential function with a slight twist—it's in the form of decay because of the negative exponent \( -t^4 \). This means our function \( y(t) \) decreases as \( t \) increases, much like the glow of fireflies fading into the night. Understanding the nature of the exponential function was crucial in the step-by-step solution; it provided us with insight into how the function behaves and what its derivative should look like, leading us to successfully verify the solution of the differential equation.