The equation \(y^2=cx\) is the general solution of:
A\(y'=\frac{2y}{x}\)
B\(y'=\frac{2x}{y}\)
C\(y'=\frac{y}{2x}\)
D\(y'=\frac{x}{2y}\)
Mathematics · Question Bank
Differential Equations
39 of 42 questions match · page 1 of 2. Answer in your head first, then check the solution.
Solve the differential equation: \(x(y-1)dx+(x+1)dy=0\). If \(y=2\) when \(x=1\), determine \(y\) when \(x=2\).
A1.80
B1.48
C1.55
D1.63
If \(dy=x^2 dx\); what is the equation of \(y\) in terms of \(x\) if the curve passes through \((1,1)\)?
A\(x^3-3y+3=0\)
B\(x^3-3y+2=0\)
C\(x^3+3y^2+2=0\)
D\(2y+x^3+2=0\)
Find the equation of the curve at every point of which the tangent line has a slope of \(2x\).
A\(x=-y^2+C\)
B\(y=-x^2+C\)
C\(y=x^2+C\)
D\(x=y^2+C\)
Solve \((\cos x \cos y-\cot x)dx-\sin x \sin y\,dy=0\).
A\(\sin x \cos y=\ln(c \cos x)\)
B\(\sin x \cos y=\ln(c \sin x)\)
C\(\sin x \cos y=-\ln(c \sin x)\)
D\(\sin x \cos y=-\ln(c \cos x)\)
Solve the differential equation \(dy-x\,dx=0\), if the curve passes through \((1,0)\)?
A\(3x^2+2y-3=0\)
B\(2y+x^2-1=0\)
C\(x^2-2y-1=0\)
D\(2x^2+2y-2=0\)
What is the solution of the first order differential equation \(y(k+1)=y(k)+5\)?
A\(y(k)=4-\frac{5}{k}\)
B\(y(k)=20+5k\)
C\(y(k)=C-k\), where C is constant
DThe solution is non-existent for real values of y
Solve \((y-\sqrt{x^2+y^2})dx-x\,dy=0\).
A\(\sqrt{x^2+y^2}+y=C\)
B\(\sqrt{x^2+y^2+y}=C\)
C\(\sqrt{x+y}+y=C\)
D\(\sqrt{x^2-y}+y=C\)
Find the differential equation whose general solution is \(y=C_1 x + C_2 e^x\).
A\((x-1)y''-xy'+y=0\)
B\((x+1)y''-xy'+y=0\)
C\((x-1)y''+xy'+y=0\)
D\((x+1)y''+xy'+y=0\)
Find the general solution of \(y'=y \sec x\).
A\(y=C(\sec x+\tan x)\)
B\(y=C(\sec x-\tan x)\)
C\(y=C \sec x \tan x\)
D\(y=C(\sec^2 x \tan x)\)
Solve \(xy'(2y-1)=y(1-x)\).
A\(\ln(xy)=2(x-y)+C\)
B\(\ln(xy)=x-2y+C\)
C\(\ln(xy)=2y-x+C\)
D\(\ln(xy)=x+2y+C\)
Solve \((x+y)dy=(x-y)dx\).
A\(x^2+y^2=C\)
B\(x^2+2xy+y^2=C\)
C\(x^2-2xy-y^2=C\)
D\(x^2-2xy+y^2=C\)
Solve the linear equation: \(\frac{dy}{dx}+\frac{y}{x}=x^2\).
A\(xy^2=\frac{x^3}{4}+C\)
B\(xy=\frac{x^4}{4}+C\)
C\(x^2y=\frac{x^4}{4}+C\)
D\(y=\frac{x^3}{4}+C\)
Find the differential equations of the family of lines passing through the origin.
A\(y\,dx-x\,dy=0\)
B\(x\,dy-y\,dx=0\)
C\(x\,dx+y\,dy=0\)
D\(y\,dx+x\,dy=0\)
What is the differential equation of the family of parabolas having their vertices at the origin and their foci on the x-axis.
A\(2x\,dy-y\,dx=0\)
B\(x\,dy+y\,dx=0\)
C\(2y\,dx-x\,dy=0\)
D\(\frac{dy}{dx}-x=0\)
Determine the differential equation of the family of lines passing through \((h,k)\).
A\((y-k)dx-(x-h)dy=0\)
B\((y-h)+(y-k)=\frac{dy}{dx}\)
C\((x-h)dx-(y-k)dy=0\)
D\((x+h)dx-(y-k)dy=0\)
Determine the differential equation of the family of circles with center on the y-axis.
A\((y'')^3-xy''+y'=0\)
B\(y''-xyy'=0\)
C\(xy''-(y')^3-y'=0\)
D\((y')^3+(y'')^2+xy=0\)
Radium decomposes at a rate proportional to the amount at any instant. In 100 years, 100 mg of radium decomposes to 96 mg. How many mg will be left after 100 years?
A88.60
B95.32
C92.16
D90.72
The population of a country doubles in 50 years. How many years will it be five times as much? Assume that the rate of increase is proportional to the number of inhabitants.
A100 years
B116 years
C120 years
D98 years
Radium decomposes at a rate proportional to the amount present. If half of the original amount disappears after 1000 years, what is the percentage lost in 100 years?
A6.70%
B4.50%
C5.36%
D4.30%