Question

Solving initial value problems Determine whether the following equations are separable. If so, solve the initial value problem. $$y^{\prime}(t)=y\left(4 t^{3}+1\right), y(0)=4$$

Short answer

Question: Determine the solution to the given initial value problem: $$y'(t) = y(4t^3 + 1), y(0)=4$$ Answer: The solution to the given initial value problem is $$y(t) = e^{t^4 + t + \ln(4)}$$.

Step-by-step solution

Determine if the equation is separable

To check if the equation is separable, we rewrite the equation in terms of y and t. The given equation is: $$y'(t) = y(4t^3 + 1)$$ Let's try to rewrite it separating the variables by dividing by y and differentiate with respect to t: $$\frac{dy}{dt} = y(4t^3 + 1)$$ So, we have: $$\frac{dy}{y} = (4t^3+1) dt$$ Here, we have succeeded in separating the variables, which means the equation is separable.

Integrate both sides

Now, to solve the initial value problem, we need to integrate both sides of the equation. We do this as follows: $$\int \frac{dy}{y} = \int (4t^3+1) dt$$ On integrating, we get: $$\ln|y| = \int (4t^3+1) dt$$ Now integrate the right-hand side: $$\ln|y| = t^4 + t + C_1$$ To solve for y, we need to use the exponent function: $$y(t) = e^{t^4 + t + C_1}$$

Apply the given initial condition

Now, we have to apply the given initial condition, which is \(y(0) = 4\). By plugging in the values, we get: $$4 = e^{0^4 + 0 + C_1}$$ $$4 = e^{C_1}$$ Now, to find the value of \(C_1\), we take the natural logarithm: $$C_1 = \ln(4)$$

Write the final answer

We now substitute the value of \(C_1\) back into the equation \(y(t)\) to get the final answer: $$y(t) = e^{t^4 + t + \ln(4)}$$ This is the solution to the given initial value problem.

Key concepts

Separable Differential Equations

Understanding separable differential equations is key to solving many types of initial value problems. A differential equation is said to be separable if it can be expressed in the form \( \frac{dy}{dx} = g(x)h(y) \), where the function on the right side of the equation is the product of two functions, each dependent on a single variable. In other words, the variables can be separated and then integrated separately.

The process generally involves rearranging the equation to isolate \( dy \) and \( dx \) on different sides. For example, the given equation \( y'(t) = y(4t^3 + 1) \) is transformed into \( \frac{dy}{y} = (4t^3+1) dt \), thereby separating the variables and setting up for integration. It is important to remember that integrals can include an arbitrary constant, which will be determined later by applying the initial conditions.

Integration of Functions

Integration is an essential operation in calculus, often thought of as the reverse process of differentiation, and is used for obtaining the function from its rate of change. In the context of separable differential equations, after separating the variables, you need to integrate both sides to find a general solution.

The integral \( \int \frac{dy}{y} \) represents a common form that you’ll see, resulting in the natural logarithm of the absolute value of \( y \). Similarly, the integration of a function of \( t \) on the right-hand side will vary depending on the form of the function. These integrations may involve simple antiderivatives for polynomials, or more complex integrals for other types of functions.

Applying Initial Conditions

After solving the indefinite integrals on both sides of a separable equation, we get a general solution that includes arbitrary constants. To find the specific solution that satisfies the initial value problem, we apply the initial conditions given in the problem.

In this context, the initial condition \( y(0) = 4 \) provides a specific point on the curve described by the differential equation. By substituting this condition into the general solution, we can solve for the constant \( C_1 \). It is crucial to ensure that the initial condition is applied correctly to find an exact solution that represents the physical or mathematical scenario being modeled.

Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is an irrational and transcendental constant approximately equal to 2.71828. This function is the inverse of the exponential function and arises naturally in the process of integrating the function \( 1/x \).

In the given exercise, the natural logarithm is used to find the value of the constant \( C_1 \) when the initial condition is applied. With \( \ln(4) \) representing \( C_1 \) in our problem, it allows us to express the solution to the initial value problem with a specific constant, rather than a general one. In calculus and differential equations, understanding the relationship between the exponential function and the natural logarithm is pivotal for solving various types of equations.