Example 1: Initial Value Problem

Given the differential equation: $f\prime \left(x\right)= x^3+{5x}^2–7x+3$

Determine its specific solution given (6,586).

Step 1: Find your general solution using your standard Indefinite Integral antiderivative process.

Her you will find the antiderivative of the given equation, $f\prime \left(x\right)= x^3+{5x}^2–7x+3$, using your standard antiderivative methods. This problem will only require the use of the power rule and the constant rule.

$\int x^3+{5x}^2–7x+3 \mathit{dx}$

$\int x^3+{5x}^2–7x^1+3 \mathit{dx}$

$=\frac{x^{3+1}}{4}+\frac{5x^{2+1}}{3}–\frac{7x^{1+1}}{2}+3x\mathit{+}\mathit{C}$

$\mathit{=}\frac{1}{4}{x}^4+\frac{5}{3}{x}^3–\frac{7}{2}{x}^2+3x\mathit{+}\mathit{C}$

$f\left(x\right)=\frac{1}{4}{x}^4+\frac{5}{3}{x}^3–\frac{7}{2}{x}^2+3x\mathit{+}\mathit{C}$

Step 2: Plug your given point ( x ,y) into your general solution from Step 1 and solve for the +C value.

Here you are given the point (6,586). You will replace all the x ’s with 6 . You will replace the $f\left(x\right)$ with 586. Remember $f\left(x\right)=y$.

Simplify the right side of the equals where all the x ’s were plugged in. You will then have a very straightforward algebra problem you will need to solve for C .

$f\left(x\right)=\frac{1}{4}{x}^4+\frac{5}{3}{x}^3–\frac{7}{2}{x}^2+3x\mathit{+}\mathit{C}$

$586=\frac{1}{4}{\left(6\right)}^4+\frac{5}{3}{\left(6\right)}^3–\frac{7}{2}{\left(6\right)}^2+3\left(6\right)\mathit{+}\mathit{C}$

$586=324+360–126+18\mathit{+}\mathit{C}$

$586=576\mathit{+}\mathit{C}$

$\begin{array}{c}586\\ –576\end{array}=\begin{array}{c}576\\ –576\end{array}\mathit{+}\mathit{C}$

$10=\mathit{C}$

Step 3: Plug the C-value you found in Step 2 into the general solution from Step 1 to get the final result a specific solution.

$f\left(x\right)=\frac{1}{4}{x}^4+\frac{5}{3}{x}^3–\frac{7}{2}{x}^2+3x\mathit{+}\mathit{C}$

$f\left(x\right)=\frac{1}{4}{x}^4+\frac{5}{3}{x}^3–\frac{7}{2}{x}^2+3x\mathit{+}\mathit{10}$

Final Result Meaning:

The derivative, $f\prime \left(x\right)= x^3+{5x}^2–7x+3$, whose antiderivative, Indefinite Integral, general solution is $f\left(x\right)=\frac{1}{4}{x}^4+\frac{5}{3}{x}^3–\frac{7}{2}{x}^2+3x\mathit{+}\mathit{C}$, has the specific solution, $f\left(x\right)=\frac{1}{4}{x}^4+\frac{5}{3}{x}^3–\frac{7}{2}{x}^2+3x\mathit{+}\mathit{10}$, that includes the given initial value (6,586).