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Plot the isoclines of the equation y = xy′ + y′′
If is the solution of a differential equation, the points at which has a constant value lie on a line for each . This line is called an isocline.
Plot some isoclines and construct a direction field of the equation
Determine a one-parameter family of solutions of this equation from the appearance of the direction field.
Incomplete.
Integral Curves
Ordinary Differential Equations
Show that the isoclines of y′ = x + y form a one-parameter family of lines
Show that the isoclines of the differential equation
form a one-parameter family of straight lines. Make a plot of the isoclines corresponding to the slopes . Using these isoclines, construct a direction field and sketch the integral curve passing through the origin. Identify one of the integral curves is also an isocline.
The isoclines of are the curves which implies . These are straight lines with slope . The integral curve passing through the origin is . The isocline is also an integral curve.
Plot the isoclines of the differential equation y′ = x2 + y2
For the differential equation
plot the isoclines corresponding to the constant slopes , and 2. Using these isoclines, construct a direction field for the equation and determine the shape of the integral curve which passes through the origin.
The isoclines of the differential equation are concentric circles centered at the origin with slope equal to the radius of the circle. The shape of the integral curve passing through the origin is cubic.
Find a first-order differential equation whose integral curves are all circles through (1,1) and (-1,-1)
Find a first-order differential equation whose integral curves consist of all circles through the points and .
Circles going through both the points and must have their center on the line , say at . The radius is given by
Therefore we have the equation
Differentiating both sides with respect to ,
From the original equation we can solve for the constant,
Find a first-order differential equation having all circles through (1,0) and (-1,0) as integral curves
Find a first-order differential equation with all circles through the points and as integral curves.
Circles that go through both the points and must have center on the -axis, at say. Then the radius is given by
Therefore, they all satisfy the equation
Differentiating both sides of this equation with respect to we have,
From the original equation we can also solve for the constant,
Therefore we have,
Find a first-order differential equation having the family arctan y + arcsin x = C as integral curves
Find a first-order differential equation having the family
as integral curves.
We differentiate both sides of the given equation with respect to ,
This is a first-order differential equation with the given family of curves as integral curves.
Find a first-order differential equation having the family y = c (cos x) as integral curves
First, we differentiate both sides of the given equation with respect to ,
Find a first-order differential equation having the family y4 (x + 2) = C(x – 2) as integral curves
Find a first-order differential equation having the family y = C(x-1)ex as integral curves
Starting with the given equation we differentiate both sides with respect to ,
From the original equation we can solve for ,
Therefore, we have the first-order differential equation
having the given family as integral curves.
Find a first-order differential equation having the family x2 + y2 + 2Cy = 1 as integral curves
From the given we equation we can solve for the constant,
Therefore we have
This is a first-order differential equation have the given family as integral curves.