Table of contents

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

Previous problemNext problem

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Which of the following is the solution to the differential equation $\frac{d y}{d x} = 4 x y$ , where $y (2) = - 2$ ?

y = - 2 e^{2 (x^{2} - 4)}

y = - 2 x^{4}

y = - 2 e^{2 (x^{2} - 2^{2})}

y = - 2 e^{4 x^{2}}

Previous problemNext problem

Verified step by step guidance

Step 1: Recognize that the given differential equation is dy/dx = 4x y, which is a first-order linear differential equation. This type of equation can often be solved using the method of separation of variables.

Step 2: Rewrite the equation to separate the variables y and x. Divide both sides by y and multiply both sides by dx to get (1/y) dy = 4x dx.

Step 3: Integrate both sides of the equation. The left-hand side becomes ∫(1/y) dy, which is ln|y|, and the right-hand side becomes ∫4x dx, which is 2x^2. After integration, the equation becomes ln|y| = 2x^2 + C, where C is the constant of integration.

Step 4: Solve for y by exponentiating both sides to remove the natural logarithm. This gives y = e^(2x^2 + C). Since e^C is just another constant, we can rewrite this as y = Ce^(2x^2), where C is a constant.

Step 5: Use the initial condition y(2) = -2 to solve for the constant C. Substitute x = 2 and y = -2 into the equation y = Ce^(2x^2). This gives -2 = Ce^(2(2^2)), or -2 = Ce^(8). Solve for C to find its value, and substitute it back into the equation to get the final solution.