11.1.2 Solving Separable Differential Equations

Question 1

Solving Separable Differential Equations

Answer 1

• To solve a separable differential equation, collect the x-terms with the dx differential and the y-terms with the dy differential.
• Initial conditions allow you to find a particular solution to a differential equation.

Question 2

note

Answer 2

• Check whether this differential equation is separable.
  Multiply both sides of the equation by dx.
• If you can gather all the x-terms with dx and all the y-terms with dy, then the differential equation is separable. You can then integrate each side of the equation.
• Each integral will have its own constant of integration, but you can combine them into a single constant C.
• This general solution can even be solved for y.
• Varying the value of the constant produces multiple solutions. You can graph them to form the solution curves.
• If you know the value of y corresponding to a given value of x, then you use can use this initial condition to determine a particular solution.
• When you plug in the values of x and y, the only unknown is C.
• Solve for C and plug it into the general solution to produce a particular solution.
• Multiply both sides of this differential equation by dx to
  determine whether it is separable.
• It is separable, so integrate both sides.
• The two constants of integration combine to form one constant in the general solution. Varying the value of C produces a family of solution curves.
• General solutions do not always have to be solved for y.

Question 3

Which of the following is a solution of dy/dx=y^2?

Answer 3

y=−1/x+1

Question 4

Which of the following is the general solution of dy/dx=xe^x+y ?

Answer 4

–e^−y=xex−ex+C

Question 5

Which of the following is the general solution of dydx=x/y^2+1 ?

Answer 5

x^2/2=y^3/3+y+C1

Question 6

Which of the following is the general solution of 1/x^2+x dy/dx=y^−2?

Answer 6

y^3/3=x^3/3+x^2/2+C

Question 7

Which of the following is the general solution of y2+y2x2+xdydx=1?

Answer 7

2x33+x22=y33+y22+C

Question 8

Which of the following is the general solution of dydx=xy2y4+1 ?

Answer 8

x22=y33−1y+C

Question 9

Which of the following is the general solution of dydx=xy?

Answer 9

y 2 = x 2 + C

Question 10

Which of the following is the general solution of dydx=y2(x3+x2+1)?

Answer 10

−1y=x44+x33+x+C

Question 11

Which of the following is the general solution of dydx=2xy2?

Answer 11

y=−1x2+C

Question 12

Which of the following is the general solution of dydx=sinxy+1?

Answer 12

y22+y=C−cosx

Question 13

Which of the following is the general solution of dydx=xy2−y2+x−1?

Answer 13

arctany=x22−x+C