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Solve the differential equation
It is given that
Differentiating it w.r.t. y, we get,
From equation (1) and equation (2), we get,
⇒ dz = dy
On integrating both sides, we get,
z = y + C
For each of the differential equations given below, indicate its order and degree (if defined).
For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
xy = a ex + b e–x + x2 :
y = ex (a cos x + b sin x) :
y = x sin 3x :
x2 = 2y2 log y :
Form the differential equation representing the family of curves given by (x – a)2 + 2y2 = a2, where a is an arbitrary constant.
Prove that x2 – y2 = c (x2 + y2)2 is the general solution of differential equation (x3–3xy2) dx = (y3–3x2y)dy, where c is a parameter.
Form the differential equation of the family of circles in the first quadrant which touch the coordinate axes.
Find the general solution of the differential equation
Show that the general solution of the differential equation is given by (x + y + 1) = A (1 – x – y – 2xy), where A is parameter.
Find the equation of the curve passing through the point whose differential equation is sin x cos y dx + cos x sin y dy = 0.
Find the particular solution of the differential equation (1 + e2x) dy + (1 + y2) ex dx = 0, given that y = 1 when x = 0.
Find a particular solution of the differential equation (x – y) (dx + dy) = dx – dy, given that y = –1, when x = 0. (Hint: put x – y = t)
Find a particular solution of the differential equation (x ≠ 0), given that y = 0 when
Find a particular solution of the differential equation , given that y = 0 when x = 0.
The population of a village increases continuously at the rate proportional to the number of its inhabitants present at any time. If the population of the village was 20, 000 in 1999 and 25000 in the year 2004, what will be the population of the village in 2009?
The general solution of the differential equation is
The general solution of a differential equation of the type is
The general solution of the differential equation ex dy + (y ex + 2x) dx = 0 is