Differential Equation: Separation of variables. Show a complete solution.Solve the differential equation using Separation of variables:

Zoe Oneal

Zoe Oneal

Answered question

2021-09-09

Differential Equation: Separation of variables. Show a complete solution.
Solve the differential equation using Separation of variables:
1y2dx1x2dy=0.y(0)=32

Answer & Explanation

Caren

Caren

Skilled2021-09-10Added 96 answers

Step 1
Here we solve the differential equation using separation of variables
1y2dx1x2dy=0.y(0)=32
Step 2
Now 1y2dx1x2dy=0
1y21x2dydx=0
dydx=1y21x2
Next we proceed by way of separation of variables.
dydx=1y21x2
dy1y2=dx1x2
Integrating dy1y2=dx1x2+C
sin1(y)=sin1(x)+C(1)
Finally we use the initial condition to solve the constant C. y(0)=32x=0,y=32
from(1)
sin1(32)=sin10+C
π3=0+C
C=π3
Thus the solution is given by
sin1y=sin1x+π3
Solving for y the explicit solution
y=sin(sin1x+π3)

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