57
MCQ (Single Correct Answer)
The solution of $x \frac{d y}{d x}+y=e^x$ is
A
$y=\frac{\mathrm{e}^x}{x}+\frac{k}{x}$
B
$y=x e^x+C x$
C
$y=x e^x+k$
D
$x=\frac{e^y}{y}+\frac{k}{y}$
58
MCQ (Single Correct Answer)
The differential equation of the family of curves $x^2+y^2-2 a y=0$, where $a$ is arbitrary constant, is
A
$\left(x^2-y^2\right) \frac{d y}{d x}=2 x y$
B
$2\left(x^2+y^2\right) \frac{d y}{d x}=x y$
C
$2\left(x^2-y^2\right) \frac{d y}{d x}=x y$
D
$\left(x^2+y^2\right) \frac{d y}{d x}=2 x y$
59
MCQ (Single Correct Answer)
The family $Y=A x+A^3$ of curves will correspond to a differential equation of order
A
3
B
2
C
1
D
not defined
60
MCQ (Single Correct Answer)
The general solution of $\frac{d y}{d x}=2 x e^{x^2-y}$ is
A
$\mathrm{e}^{x^2-y}=C$
B
$\mathrm{e}^{-y}+\mathrm{e}^{x^2}=C$
C
$\mathrm{e}^y=\mathrm{e}^{x^2}+C$
D
$\mathrm{e}^{x^2+y}=C$
61
MCQ (Single Correct Answer)
The curve for which the slope of the tangent at any point is equal to the ratio of the abcissa to the ordinate of the point is
A
an ellipse
B
parabola
C
circle
D
rectangular hyperbola