Verify that the given function is a solution of the corresponding differential equation:
Given: \(y = e^x + 1\)
Differential equation: \(y'' - y' = 0\)
Verify that the given function is a solution of the corresponding differential equation:
Given: \(y = x^2 + 2x + C\)
Differential equation: \(y' - 2x - 2 = 0\)
Verify that the given function is a solution of the corresponding differential equation:
Given: \(y = \cos x + C\)
Differential equation: \(y' + \sin x = 0\)
Verify that the given function is a solution of the corresponding differential equation:
Given: \(y = \sqrt{1 + x^2}\)
Differential equation: \(y' = \frac{xy}{1 + x^2}\)
Verify that the given function is a solution of the corresponding differential equation:
Given: \(y = Ax\)
Differential equation: \(xy' = y\) \((x \ne 0)\)
Verify that the given function is a solution of the corresponding differential equation:
Given: \(y = x\sin x\)
Differential equation: \(xy' = y + x\sqrt{x^2 - y^2}\) \((x \ne 0\ \text{and}\ x > y\ \text{or}\ x < -y)\)
Verify that the given function is a solution of the corresponding differential equation:
Given: \(xy = \log y + C\)
Differential equation: \(y' = \frac{y^2}{1 - xy}\) \((xy \ne 1)\)
Verify that the given function is a solution of the corresponding differential equation:
Given: \(y - \cos y = x\)
Differential equation: \((y\sin y + \cos y + x)\,y' = y\)
Verify that the given function is a solution of the corresponding differential equation:
Given: \(x + y = \tan^{-1}y\)
Differential equation: \(y^2y' + y^2 + 1 = 0\)
Verify that the given function is a solution of the corresponding differential equation:
Given: \(y = \sqrt{a^2 - x^2},\ x \in (-a, a)\)
Differential equation: \(x + y\frac{dy}{dx} = 0\) \((y \ne 0)\)
The number of arbitrary constants in the general solution of a differential equation of fourth order are:
0
2
3
4
The number of arbitrary constants in the particular solution of a differential equation of third order are:
3
2
1
0