Q.Choose the correct answer.
Q. The initial value value problem dy/dx=|y|1/2 ,y(0)=0 has:
a) unique solution
b) No solution
c) Infinitely many solution
d) Two solutions.
1
Expert's answer
2019-07-10T13:31:30-0400
The initial value value problem dy/dx=|y|1/2 ,y(0)=0 has
d) Two solutions.
Find the general solution of
y′=∣y∣
dxdy=∣y∣
1) Let y(x)≤0 :
dxdy=−y
We can separate the variables:
−ydy=dx
−−yd(−y)=dx
Now, integrate the left-hand side dy and the right-hand side dx
−∫−yd(−y)=∫dx
−2−y=x+C1
−y=−21(x+C1)
−y=41(x+C1)2,x+C1≤0
y=−41(x+C1)2,x+C1≤0(1)
2) Let y(x)≥0 :
dxdy=∣y∣⇒dxdy=y
We can separate the variables:
ydy=dx
Now, integrate the left-hand side dy and the right-hand side dx
∫yd(y)=∫dx
2y=x+C1
y=21(x+C1)
y=41(x+C1)2,x+C1≥0(2)
So there is our general solution (from (1), (2)):
y=41(x+C1)2,x+C1≥0y=−41(x+C1)2,x+C1≤0
To find the particular solution, substitute the initial condition values to obtain
y(0)=0⇒0=41(0+C1)2,⇒C1=0
So, the particular solution that satisfies the initial condition is
y=−41x2,x≥0y=−41x2,x≤0
But the solution to equality is y=0 , which also satisfies the condition y(0)=0 .
The initial value value problem dy/dx=|y|1/2 ,y(0)=0 has Two solutions:
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