What is the solution of the differential equation \(\dfrac{dy}{dx} = 1 + x\cot(y-x)\)?
\(\sec(y-x) = ce^{x^{2}/2}\)
Let \(v=y-x\), so \(\frac{dv}{dx}=\frac{dy}{dx}-1 = x\cot v\). This separates as \(\tan v\,dv = x\,dx\). Integrating, \(-\ln|\cos v| = \frac{x^2}{2}+C\), which gives \(\sec v = ce^{x^2/2}\). Substituting back \(v=y-x\) gives \(\sec(y-x)=ce^{x^2/2}\).
Similar Questions
What is the general solution of the differential equation ydx – (x + 2y 2) dy = 0?
If xdy = y(dx + ydy) ; y(1) = 1 and y(x) > 0, then what is y(-3) equal to?
A particle starts from origin with a velocity (in m/s) given by the equation \(\rm \frac{dx}{dt}=x+1\) . The time (in seconds) taken by the particle to traverse a distance of 24 m is:
- If \(\frac{d}{d x}\left(\frac{1+x^4+x^8}{1−x^2+x^4}\right)\) = ax + bx 3 , then which one of the following is correct?
What is the solution of the differential equation (dy − dx) + cos x(dy + dx) = 0 ?
What is the solution of the following differential equation?
\(\rm \ln\left(\frac{dy}{dx}\right)+y = x\)
The solution of the differential equation \(\frac{{dy}}{{dx}} = \cos \left( {y - x} \right) + 1\) is
What is the solution of the differential equation x dy – y dx = 0?
What is the general solution of the differential equation \(\frac{{dy}}{{dx}} + \frac{x}{y} = 0\) ?
What is the solution of the differential equation \(\ln \left( {\frac{{{\rm{dy}}}}{{{\rm{dx}}}}} \right) - {\rm{a}} = 0?\)
Important Questions from Solution of Differential Equations
- The solution of the differential equation \(\rm\left(\frac{dy}{dx}\right)^2−\frac{d^2y}{dx^2}\) = e y , with the boundary conditions y(0) = 0 and y'(0) = −1, is
The equation of motion of a one-dimensional forced harmonic oscillator in the presence of a dissipative force is described by \(\frac{{{{\rm{d}}^{\rm{2}}}{\rm{x}}}}{{{\rm{d}}{{\rm{t}}^{\rm{2}}}}}\,{\rm{ + }}\,{\rm{10}}\frac{{{\rm{dx}}}}{{{\rm{dt}}}}\,{\rm{ + }}\,{\rm{16x}}\,{\rm{ = }}\,{\rm{6t}}{{\rm{e}}^{{\rm{ - 8t}}}}{\rm{ + }}\,{\rm{4}}{{\rm{t}}^{\rm{2}}}{{\rm{e}}^{{\rm{ - 2t}}}}\) The general form of the particular solution, in terms of constants A, B etc., is
What is the general solution of the differential equation ydx – (x + 2y 2) dy = 0?
If xdy = y(dx + ydy) ; y(1) = 1 and y(x) > 0, then what is y(-3) equal to?
If y(x) is a solution of the differential equation \(\frac{{dy}}{{dx}} + 4xy = {x^3},y(0) = 0\) then \(\mathop {\lim }\limits_{x \to 0} y(x)\) is
