11.1.2 Solving Separable Differential Equations Flashcards

1
Q

Solving Separable Differential Equations

A
  • 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.
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2
Q

note

A
  • 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.
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3
Q

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

A

y=−1/x+1

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4
Q

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

A

–e^−y=xex−ex+C

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5
Q

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

A

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

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6
Q

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

A

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

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7
Q

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

A

2x33+x22=y33+y22+C

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8
Q

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

A

x22=y33−1y+C

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9
Q

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

A

y 2 = x 2 + C

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10
Q

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

A

−1y=x44+x33+x+C

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11
Q

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

A

y=−1x2+C

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12
Q

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

A

y22+y=C−cosx

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13
Q

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

A

arctany=x22−x+C

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