Modeling by Solving Separable Differential Equations - AP Calculus BC

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Question

Solve the separable differential equation

$\frac{\mathrm{d} y}{\mathrm{d} x}= yx^2$

with the condition .

Answer

To solve the separable differential equation, we must separate x and y, dx and dy respectively to opposite sides:

$\frac{dy}{y}=x^2dx$

Integrating both sides, we get

$\ln\left | y\right |=\frac{ x^3}{3}+C$

The rules of integration used were

$\int \frac{dx}{x}=\ln\left | x\right |+C$ , $\int x^ndx= \frac{x^{n+1}}{n+1}+C$

The constants of integration merged into one.

Now, we exponentiate both sides of the equation to solve for y, and use the properties of exponents to simplify:

$y=e^{\frac{x^3}{3}}+e^C=e^{{\frac{x^3}{3}}\cdot C}=Ce^{\frac{x^3}{3}}$

To solve for C, we use our given condition:

Our final answer is

$y=4e^{\frac{x^3}{3}}$

Compare your answer with the correct one above

Question

Solve the separable differential equation

$\frac{\mathrm{d} y}{\mathrm{d} x}= yx^2$

with the condition .

Answer

To solve the separable differential equation, we must separate x and y, dx and dy respectively to opposite sides:

$\frac{dy}{y}=x^2dx$

Integrating both sides, we get

$\ln\left | y\right |=\frac{ x^3}{3}+C$

The rules of integration used were

$\int \frac{dx}{x}=\ln\left | x\right |+C$ , $\int x^ndx= \frac{x^{n+1}}{n+1}+C$

The constants of integration merged into one.

Now, we exponentiate both sides of the equation to solve for y, and use the properties of exponents to simplify:

$y=e^{\frac{x^3}{3}}+e^C=e^{{\frac{x^3}{3}}\cdot C}=Ce^{\frac{x^3}{3}}$

To solve for C, we use our given condition:

Our final answer is

$y=4e^{\frac{x^3}{3}}$

Compare your answer with the correct one above

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