11.1.2 Solving Separable Differential Equations Flashcards
Solving Separable Differential Equations
- 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.
note
- 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.
Which of the following is a solution of dy/dx=y^2?
y=−1/x+1
Which of the following is the general solution of dy/dx=xe^x+y ?
–e^−y=xex−ex+C
Which of the following is the general solution of dydx=x/y^2+1 ?
x^2/2=y^3/3+y+C1
Which of the following is the general solution of 1/x^2+x dy/dx=y^−2?
y^3/3=x^3/3+x^2/2+C
Which of the following is the general solution of y2+y2x2+xdydx=1?
2x33+x22=y33+y22+C
Which of the following is the general solution of dydx=xy2y4+1 ?
x22=y33−1y+C
Which of the following is the general solution of dydx=xy?
y 2 = x 2 + C
Which of the following is the general solution of dydx=y2(x3+x2+1)?
−1y=x44+x33+x+C
Which of the following is the general solution of dydx=2xy2?
y=−1x2+C
Which of the following is the general solution of dydx=sinxy+1?
y22+y=C−cosx
Which of the following is the general solution of dydx=xy2−y2+x−1?
arctany=x22−x+C