# Difference between unique solution and particular solution

by inter060708
Tags: difference, solution, unique
 P: 21 For first order ordinary differential equations. Are each particular solution a subset of unique solutions? Please explain, thanks
 P: 350 A particular solution of an equation is a specific function that satisfies the equation. e.g. y' = y. A particular solution is y = 2e^x. Uniqueness is a word that means that there is only one. In the context of Intro to Diff. Eq. it is used in connection with initial value problems. e.g. y'=y, y(0) = 5. There is only one solution of this initial value problem: y=5e^x. We say that this problem has a unique solution. Consider $y'=(3/2)y^{1/3}, y(0)=0$. y=0 (the constant function) is a solution of this problem. y=x^(3/2) is another solution of this problem. This problem does not have a unique solution. Consider (y')^2+y^2=1 with initial value y(0)=1. Then y = 1 (the constant function) and y=cos(x) are two solutions. Therefore this problem also does not have a unique solution.
Math
Emeritus
For example, the differential equation y''= y has "general solution" $y(x)= Ae^x+ Be^{-x}$. That is, any choice of values for A and B give a solution and every solution is of that form. If we choose specific values for A and B, say arbitrarily select A= 1, B= 2, we have $y(x)= e^x+ 2e^{-x}$ which is a "specific" solution.
If, instead, we have the "initial value problem" with differential equation y''= y and the conditions that y(0)= 1, y'(0)= 0, then we have a unique solution- it must be of the form $y(x)= Ae^x+ Be^{-x}$, as must all solutions, and we must have $y(0)= Ae^0+ Be^0= A+ B= 1$ and $y'(0)= Ae^0- Be^0= A- B= 0$. Those two equations are satisfied by A= B= 1/2. That is, the unique solution to this problem is $y(x)= (1/2)e^x+ (1/2)e^{-x}$.