Determine the solution to the initial value differential equation:
y′+2*y^2*e^(5*x)= 0
y(0)=2
To determine the solution to the initial value differential equation, we can use the method of separation of variables.
Step 1: Rewrite the equation in a suitable form for separation of variables.
Start with the given differential equation:
y' + 2y^2e^(5x) = 0
Step 2: Separate the variables.
Move all the terms involving y to one side and all the terms involving x to the other side:
dy/y^2 = -2e^(5x)dx
Step 3: Integrate both sides of the equation.
Integrate both sides separately:
∫(dy/y^2) = ∫(-2e^(5x)dx)
The integral of (dy/y^2) can be rewritten as -1/y. For the integral of (-2e^(5x)dx), use the substitution u = 5x, du = 5dx:
-1/y = ∫(-2e^(5x)dx)
-1/y = -2/5 ∫e^u du
-1/y = -2/5 * e^u + C
where C is the constant of integration.
Step 4: Solve for y.
Multiply both sides of the equation by y to isolate y in the denominator:
-1 = (-2/5 * e^u + C) * y
Now substitute back u = 5x:
-1 = (-2/5 * e^(5x) + C) * y
Finally, substitute the initial value y(0) = 2:
-1 = (-2/5 * e^(5*0) + C) * 2
-1 = (-2/5 * e^0 + C) * 2
-1 = (-2/5 * 1 + C) * 2
-1 = (-2/5 + C) * 2
-1 = (-4/5 + 2C)
Solve for C:
-1 + 4/5 = -4/5 + 2C
-1/5 = 2C
C = -1/10
Substitute C back into the equation:
-1 = (-2/5 * e^(5x) - 1/10) * 2
Simplify the equation:
-1 = -4/5 * e^(5x) - 1/5
Now solve for y:
y = (-1)/(-4/5 * e^(5x) - 1/5)
y = 5/(4e^(5x) + 1)
Therefore, the solution to the initial value differential equation is:
y = 5/(4e^(5x) + 1)