Find the general solution to the differential equation dydx=−2x+5x2\frac{dy}{dx}=-2x+5x^2.

y = − x 2 + 5 3 x 3 + C y=-x^2+\frac53x^3+C

Find the general solution to the differential equation d y d x = − 2 x + 5 x 2 \frac{dy}{dx}=-2x+5x^2 .

Here we're asked to find the general solution to the differential equation dy dx equals negative two x plus five x squared. So if dy dx is equal to that negative two x plus five x squared, we can find what Y is by integrating. So Y is going to be equal to the integral of negative two X plus five X squared DX. So let's go ahead and find this integral. Now for that first term here, we wanna apply our power rule, adding one to that exponent, dividing by that exponent, dividing by that new exponent. So that's going to give us here negative x squared. Then for the second term, we're doing the same thing, applying our power rule. So adding one to that exponent, dividing by our new exponent gives us five thirds x cubed. Then we wanna make sure that we add on our constant of integration here since this is an indefinite integral. So y is then equal to negative x squared plus five thirds x cubed plus c, and this is my general solution. Thanks for watching, and I'll see you in the next one.

Find the general solution to the differential equation d y d x = − 2 x + 5 x 2 \frac{dy}{dx}=-2x+5x^2 .

y = − x 2 + 5 3 x 3 + C y=-x^2+\frac53x^3+C

y = − x 2 + 10 x + C y=-x^2+10x+C

y = − x + 5 3 x 2 + C y=-x^{}+\frac53x^2+C

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

13. Intro to Differential Equations

Basics of Differential Equations

Calculus

13. Intro to Differential Equations

Basics of Differential Equations

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Multiple Choice

Find the general solution to the differential equation $\frac{d y}{d x} = - 2 x + 5 x^{2}$ .

$y = - x^{2} + \frac{5}{3} x^{3} + C$

$y = - x^{2} + 10 x + C$

$y = - x + 5 x + C$

$y = - x^{} + \frac{5}{3} x^{2} + C$

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Verified step by step guidance

Step 1: Recognize that the given equation $ \frac{dy}{dx} = -2x + 5x^2 $ is a first-order ordinary differential equation. To solve it, we will integrate both sides with respect to $ x $.

Step 2: Write the equation in integral form: $ y = \int (-2x + 5x^2) \, dx + C $, where $ C $ is the constant of integration.

Step 3: Break the integral into two separate terms: $ y = \int -2x \, dx + \int 5x^2 \, dx + C $.

Step 4: Solve each integral term-by-term: $ \int -2x \, dx = -x^2 $ and $ \int 5x^2 \, dx = \frac{5}{3}x^3 $.

Step 5: Combine the results to write the general solution: $ y = -x^2 + \frac{5}{3}x^3 + C $, where $ C $ is the constant of integration.

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