List of practice Questions

Which of the following ordinary differential equations is/are linear?

A first-order ordinary differential equation is given as follows: \[\frac{dy}{dx} + x^2 y = 0\] Which ONE of the following options CORRECTLY represents the characteristics of this equation?

The general solution to the ordinary differential equation \( \dfrac{d^2y}{dx^2} = \cos 2x \) is

If the roots of the characteristic equation of the Euler's ODE has a repeated root m, then what is the correct form of the general solution? (\(c_1,c_2\) are arbitrary constants.)

What does the differential equation \((2x+y+1)dx+(x+2y+1)dy=0\) represent?

If the non-homogeneous term in an ordinary differential equation (ODE) is \(xe^{3x}\) and 3 is a root of the characteristic equation with multiplicity 2, then what is the form of the particular solution (i.e. \(y_p\))? (c is a constant.)

The Bernoulli equation \(\frac{dy}{dx}+P(x)y=Q(x)y^3\) is best solved by making which of the following substitutions?

The general solution of the ODE \((D^2+6D+9)y=\frac{e^{-3x}}{x^3}\), \(D=\frac{d}{dx}\) (\(c_1,c_2\) are arbitrary constants) is ____.

Which one of the following options is true for the initial value problem: \(\frac{dy}{dx}=2y^{1/3}\), \(y(0)=0\)?

If \(u=x^3\) and \(v=y^2\) transforms the ODE: \(3x^5dx - y(y^2-x^3)dy=0\) into \(\frac{dy}{du}=\frac{\lambda u}{(u-v)}\), then what is the value of \(\lambda\)?