Question

Find the solution of the following initial value problems. $$p^{\prime }(t)=10e^t+70;p(0)=100$$

Short answer

Question: Find the particular solution to the given initial value problem involving a first-order differential equation: $p^{\prime }(t)=10e^t+70$, with the initial condition $p(0)=100$. Answer: The particular solution is $p(t)=10e^t+70t+90$.

Step-by-step solution

Find the general solution to the differential equation

Integrate both sides of the given differential equation with respect to $t$: $$\int p^{\prime }(t)dt=\int (10e^t+70)dt$$ This gives: $$p(t)=10\int e^{t}dt+70\int 1dt$$ Integrating gives: $$p(t)=10e^t+70t+C$$ This is the general solution to the differential equation.

Use the initial condition to find the particular solution

Now, we need to find the value of the constant $C$ given the initial condition $p(0)=100$. Substitute the initial condition into the general solution we obtained in step 1: $$100=10e^0+70(0)+C$$ $$100=10+0+C$$ Solving for $C$: $$C=90$$

Write the final particular solution

Substitute the value of the constant $C$ back into the general solution: $$p(t)=10e^t+70t+90$$ This is the required particular solution for the given initial value problem.

Key concepts

Differential Equations

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are fundamental in expressing the laws of nature and are used extensively in various fields such as physics, engineering, economics, and biology. A differential equation involves an unknown function, like $p(t)$ in our exercise, and one or more of its derivatives.

The simplest form is an ordinary differential equation (ODE) which contains only ordinary derivatives of one or more dependent variables with respect to a single independent variable. The equation $p^{\prime }(t)=10e^t+70$ from the exercise is a first-order ODE since it involves only the first derivative of the function $p(t)$ with respect to $t$.

The solution to a differential equation typically involves integration. Finding this solution, which encompasses all possible functions that satisfy the equation, is often the first step in solving many problems in science and engineering.

Integrating Factor

The integrating factor is a technique used to solve linear first-order differential equations. It multiplies the entire equation to make the left-hand side exact, that is, the derivative of a product of two functions. This method rewrites the differential equation in a form that can be integrated directly with respect to the independent variable.

For a differential equation of the form $y^{\prime }+P(x)y=Q(x)$, an integrating factor $\text{IF}$ is usually given by $e^{\int P(x)\text{dx}}$. Once the integrating factor is found, it transforms the equation into an exact differential, making it easier to find the general solution. In our initial value problem, since the equation was already given in a directly integrable form, the integrating factor wasn’t explicitly required, but the concept remains essential for understanding the solution process of more complex differential equations.

Particular Solution

A particular solution to a differential equation is a specific solution that not only satisfies the differential equation but also fulfills an initial condition or set of initial conditions. The initial condition specifies the value of the unknown function at a particular point, and it’s used to solve for any constants that appear in the general solution.

In the context of our exercise, given the initial condition $p(0)=100$, we are looking for one particular solution to the differential equation where the function $p(t)$ takes the value of 100 when $t=0$. This solution is uniquely determined by the initial condition, in contrast to the general solution, which represents a whole family of possible functions that satisfy the differential equation without considering any specific conditions.

General Solution

The general solution of a differential equation is an expression that contains all the possible solutions of the differential equation. It includes one or more arbitrary constants that represent an infinite family of solutions. The general solution incorporates every particular solution of the differential equation.

In our example, the general solution is $p(t)=10e^t+70t+C$, where $C$ is an arbitrary constant. To find a particular solution, the constant $C$ needs to be determined using the given initial conditions. Once $C$ is computed, plugging it back into the general solution will yield the particular solution $p(t)=10e^t+70t+90$ that satisfies both the differential equation and the initial condition. Understanding the general solution is key to fully grasping the structure of a differential equation’s solution space.