Solve the differential equation:
The given differential equation is: \[ 2x \frac{dy}{dx} + y(6y^2 - x - 1) = 0 \] This is a nonlinear ordinary ________ _________ _____ __________ ___ ____.
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Solve the differential equation:
Transform the given equation to Clairaut’s form and hence find its general solution:
Also find its singular solution, if it exists
Solve the differential equation:
Find the limit of
i) y = 3x
ii) y = 5x
What can you conclude about Justify your answer.
1. State whether the following statements are true or false. Give a short proof or a counterexample in support of your answer:
a) A real-valued function of three variables which is continuous everywhere is differentiable.
b) The function is not homogeneous function.
c) The cylindrical coordinates of the point whose spherical coordinates is
d) The unique solution y(x) of an ordinary differential equation:
exists
e) The differential equation is a second order differential equation of degree 3.
f) Differential equation:
in is a linear, homogeneous equation.
g) The total differential equation corresponding to the family of surfaces , where c is a parameter is .
h) Differential equation:
is a semi- linear partial differential equation of first order.
i) The differential equation:
has an integrating factor v exp(−v) .
j) The simultaneous differential equation of simple harmonic motion of a particle in phase –plane is:
Show that the differential equation:
is integrable and find its integral.
If cos and α,cosβ cos γ are the direction cosines of a line, then show that: