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
20 of 42 questions match. Answer in your head first, then check the solution.
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\(2y+x^3+2=0\)
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\(2x^2+2y-2=0\)
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 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\(y=\frac{x^3}{4}+C\)
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%
Find the equation of the family of orthogonal trajectories of the system of parabolas \(y^2=2x+c\).
A\(y=Ce^{-x}\)
B\(y=Ce^{2x}\)
C\(y=Ce^x\)
D\(y=Ce^{-2x}\)
According to Newton's law of cooling, the rate at which a substance cools in air is directly proportional to the difference between the temperature of the substance and that of air. If the temperature of the air is \(30^\circ\) and the substance cools from \(100^\circ\) to \(70^\circ\) in 15 minutes, how long will it take to cool \(100^\circ\) to \(50^\circ\)?
A33.59 min.
B43.60 min
C35.39 min
D45.30 min
If the nominal interest rate is 3%, how much is P5,000 worth in 10 years in a continuously compounded account?
AP5,750
BP6,750
CP7,500
DP6,350
A nominal interest of 3% compounded continuously is given on the account. What is the accumulated amount of P10,000 after 10 years?
AP13,620.10
BP13,500.10
CP13,650.20
DP13,498.60
Find the general solution of \((D^2 - D + 2)y = 0\).
A\(y = e^{x/2}\left(C_1\sin\tfrac{\sqrt{7}}{2}x + C_2\cos\tfrac{\sqrt{7}}{2}x\right)\)
B\(y = e^{x/2}\left(C_1\sin\tfrac{\sqrt{7}}{2}x - C_2\cos\tfrac{\sqrt{7}}{2}x\right)\)
C\(y = e^{x/2}\left(C_1\cos\tfrac{\sqrt{7}}{2}x + C_2\sin\tfrac{\sqrt{7}}{2}x\right)\)
D\(y = e^{x/2}\left(C_1\cos\tfrac{\sqrt{7}}{2}x - C_2\sin\tfrac{\sqrt{7}}{2}x\right)\)
A steel ball at 110°C cools in 8 min to 90°C in a room at 30°C. Find the temperature of the ball after 20 minutes.
A58.97°C
B68.97°C
C78.97°C
D88.97°C
What is the general solution of \((D^4 - 1)y(t) = 0\)?
A\(y = C_1e^t + C_2e^{-t} + C_3\cos t + C_4\sin t\)
B\(y = C_1e^t + C_2e^{-t} + C_3te^t + C_4te^{-t}\)
C\(y = C_1e^t + C_2e^{-t}\)
D\(y = C_1e^t + C_2te^{-t}\)
Find the general solution of \(y'' + 10y' + 41y = 0\).
A\(y = e^{-5x}(C_1\cos 4x + C_2\sin 4x)\)
B\(y = e^{5x}(C_1\cos 4x + C_2\sin 4x)\)
C\(y = e^{-4x}(C_1\cos 5x + C_2\sin 5x)\)
D\(y = e^{4x}(C_1\cos 5x + C_2\sin 5x)\)
A motorboat weighs 32,000 lb and its motor provides a thrust of 5,000 lb. Assume that the water resistance is 100 pounds for each foot per second of the speed v of the boat. Then \(1{,}000\dfrac{dv}{dt} = 5{,}000 - 100v\). If the boat starts from rest, what is the maximum velocity that it can attain?
A20 ft/s
B25 ft/s
C40 ft/s
D50 ft/s