7.7. Ordinary Differential Equations http://www.ck12.org
Exercise
- Sketch the slope field of the differential equationdydx= 1 −y. Sketch the solution curves based on it.
- Sketch the slope field of the differential equationdydx=y−x. Find the isoclines and sketch a solution curve
that passes through( 1 , 0 ).
Differential Equations and Integration
We begin the analytic solutions of differential equations with a simple type wheredy F(x,y)is a function ofxonly.
dx=f(x)is a function ofx. Then any antiderivative offis a solution by the Fundamental Theorem of Calculus:
dxd∫ax f(t)dt=f(x).
Example 1Solve the differential equationdydx=xwithy( 0 ) =1.
Solution.y=∫ x dx=x 22 +C. Theny( 0 ) =1 gives 1= 0 +C, i.e.C=1 Thereforey=x 22 +1.
Exercise
- Solve the differential equationdydx=
√
9 −x^2 withy( 0 ) =3.Solving Separable First-Order Differential Equations
The next type of differential equation where analytic solution are relatively easy is when the dependence ofF(x,y)
onxandyare separable:dydx=F(x,y)whereF(x,y) =f(x)g(y)is the product of a functions ofxandyrespectively.
The solution is in the formP(x) =Q(y). Hereg(y)is never 0 or the values ofyin the solutions will be restricted by
whereg(y) =0.
Example 1Solve the differential equationy′=xywith the initial conditiony( 0 ) =1.