Question

Find the solution of the following initial value problems. $$g^{\prime}(x)=7 x^{6}-4 x^{3}+12 ; g(1)=24$$

Short answer

Question: Solve the initial value problem (IVP) for the function $$g(x)$$ given the following first order differential equation and initial condition: $$g^{\prime}(x)=7 x^{6}-4 x^{3}+12$$ $$g(1)=24$$ Answer: The solution to the initial value problem is $$g(x) = x^{7} - x^{4} + 12x + 12$$.

Step-by-step solution

Integrate the Differential Equation

Integrate the given equation with respect to x to find the function $$g(x)$$: $$g(x) = \int (7 x^{6}-4 x^{3}+12) dx$$ Using power rule for integration and treating each term separately, we have: $$g(x) = 7 \int x^{6} dx - 4 \int x^{3} dx + 12 \int 1 dx$$ Now integrate each term: $$g(x) = 7\frac{x^{7}}{7} - 4\frac{x^{4}}{4} + 12x + C$$ Simplify the expression: $$g(x) = x^{7} - x^{4} + 12x + C$$ where C is the constant of integration.

Use the Initial Condition to Find the Constant of Integration

Now, use the initial condition $$g(1) = 24$$ to find the value of C. Substitute x=1 into the function: $$24 = (1)^7 - (1)^4 + 12(1) + C$$ Solve for C: $$24 = 1 - 1 + 12 + C$$ $$C = 24 - 12$$ $$C = 12$$

Write the IVP Solution

Now that the constant of integration is found, the solution of the initial value problem can be written. Substitute the value of C back into the function: $$g(x) = x^{7} - x^{4} + 12x + 12$$ Thus, the solution of the given initial value problem is: $$g(x) = x^{7} - x^{4} + 12x + 12$$

Key concepts

Integration of Polynomials

The process of finding the antiderivative of a polynomial function is known as the integration of polynomials. In calculus, this is a fundamental technique that allows us to solve a variety of problems, including finding the area under a curve and solving differential equations. When integrating polynomials, we apply the power rule which states that the integral of x^n, where n is any real number, is (x^(n+1))/(n+1) plus a constant of integration, denoted by C.

For example, to integrate the function 7x^6, we would add one to the exponent to get 7x^7 and then divide by the new exponent, giving us the result (7/7)x^7 or simply x^7. This rule is repeated for each term in the polynomial expression. The integration becomes slightly more complex if the polynomial involves subtraction or addition, but the underlying principle of the power rule remains the same.

Initial Condition

An initial condition is a specific detail that defines the starting point of an equation or function, and it's particularly critical when dealing with differential equations and their solutions. In the context of an initial value problem (IVP), the initial condition allows us to find the specific solution that fits the particular scenario presented. We use the initial condition to determine the constant of integration, turning a general solution with an arbitrary constant into a precise solution that satisfies the equation at the initial value. For instance, if we're given g(1) = 24 as the initial condition, it tells us the exact value of the function g(x) when x equals 1. This knowledge is pivotal because it pinpoints the exact graph or curve represented by the function out of the infinite possibilities provided by the family of solutions to a differential equation.

Constant of Integration

The constant of integration, represented as C, is an essential component in the integration process. Whenever we integrate a function, we add this constant to account for all the antiderivatives that differ by a constant amount. The presence of C denotes an entire family of curves, each shifted vertically from the others, that could potentially satisfy the given differential equation. However, when an initial condition is provided, we can determine the value of C to find the specific solution that corresponds to the exact scenario described by the problem. In the given exercise, after integrating to find g(x), we use the initial condition to calculate that C = 12. Substituting this value into the integrated function provides us with the particular solution that not only satisfies the differential equation but also the initial condition: g(x) = x^7 - x^4 + 12x + 12.