Determine order and degree (if defined) of the differential equation \(\frac{d^4y}{dx^4} + \sin(y''') = 0\).
Order 4; Degree not defined
Determine order and degree (if defined) of the differential equation \(y' + 5y = 0\).
Order 1; Degree 1
Determine order and degree (if defined) of the differential equation \(\left(\frac{ds}{dt}\right)^4 + 3s\frac{d^2s}{dt^2} = 0\).
Order 2; Degree 1
Determine order and degree (if defined) of the differential equation \(\left(\frac{d^2y}{dx^2}\right)^2 + \cos\left(\frac{dy}{dx}\right) = 0\).
Order 2; Degree not defined
Determine order and degree (if defined) of the differential equation \(\frac{d^2y}{dx^2} = \cos 3x + \sin 3x\).
Order 2; Degree 1
Determine order and degree (if defined) of the differential equation \((y''')^2 + (y'')^3 + (y')^4 + y^5 = 0\).
Order 3; Degree 2
Determine order and degree (if defined) of the differential equation \(y''' + 2y'' + y' = 0\).
Order 3; Degree 1
Determine order and degree (if defined) of the differential equation \(y' + y = e^x\).
Order 1; Degree 1
Determine order and degree (if defined) of the differential equation \(y'' + (y')^2 + 2y = 0\).
Order 2; Degree 1
Determine order and degree (if defined) of the differential equation \(y'' + 2y' + \sin y = 0\).
Order 2; Degree 1
The degree of the differential equation \(\left(\frac{d^2y}{dx^2}\right)^3 + \left(\frac{dy}{dx}\right)^2 + \sin\left(\frac{dy}{dx}\right) + 1 = 0\) is
3
2
1
not defined
The order of the differential equation \(2x^2\frac{d^2y}{dx^2} - 3\frac{dy}{dx} + y = 0\) is
2
1
0
not defined
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
For the differential equation, find the general solution:
\(\frac{dy}{dx} = \frac{1-\cos x}{1+\cos x}\)
\(y = 2\tan\frac{x}{2} - x + C\)
For the differential equation, find the general solution:
\(\frac{dy}{dx} = \sqrt{4-y^2}\) \((-2<y<2)\)
\(y = 2\sin(x + C)\)
For the differential equation, find the general solution:
\(\frac{dy}{dx} + y = 1\) \((y \ne 1)\)
\(y = 1 + Ae^{-x}\)
For the differential equation, find the general solution:
\(\sec^2 x\,\tan y\,dx + \sec^2 y\,\tan x\,dy = 0\)
\(\tan x\,\tan y = C\)
For the differential equation, find the general solution:
\((e^x + e^{-x})\,dy - (e^x - e^{-x})\,dx = 0\)
\(y = \log(e^x + e^{-x}) + C\)
For the differential equation, find the general solution:
\(\frac{dy}{dx} = (1+x^2)(1+y^2)\)
\(\tan^{-1}y = x + \frac{x^3}{3} + C\)
For the differential equation, find the general solution:
\(y\log y\,dx - x\,dy = 0\)
\(y = e^{cx}\)
For the differential equation, find the general solution:
\(x^5\frac{dy}{dx} = -y^5\)
\(x^{-4} + y^{-4} = C\)
For the differential equation, find the general solution:
\(\frac{dy}{dx} = \sin^{-1} x\)
\(y = x\sin^{-1}x + \sqrt{1-x^2} + C\)
For the differential equation, find the general solution:
\(e^x\tan y\,dx + (1-e^x)\sec^2 y\,dy = 0\)
\(\tan y = C(1 - e^x)\)
Find a particular solution satisfying the given condition:
\((x^3 + x^2 + x + 1)\frac{dy}{dx} = 2x^2 + x\), \(y = 1\) when \(x = 0\)
\(y = \frac{1}{4}\log\left[(x+1)^2(x^2+1)^3\right] - \frac{1}{2}\tan^{-1}x + 1\)
Find a particular solution satisfying the given condition:
\(x(x^2 - 1)\frac{dy}{dx} = 1\), \(y = 0\) when \(x = 2\)
\(y = \frac{1}{2}\log\left(\frac{x^2-1}{x^2}\right) - \frac{1}{2}\log\left(\frac{3}{4}\right)\)
Find a particular solution satisfying the given condition:
\(\cos\left(\frac{dy}{dx}\right) = a\) \((a \in \mathbb{R})\), \(y = 1\) when \(x = 0\)
\(\cos\left(\frac{y-2}{x}\right) = a\)
Find a particular solution satisfying the given condition:
\(\frac{dy}{dx} = y\tan x\), \(y = 1\) when \(x = 0\)
\(y = \sec x\)
Find the equation of a curve passing through the point \((0,0)\) and whose differential equation is \(y' = e^x\sin x\).
\(2y - 1 = e^x(\sin x - \cos x)\)
For the differential equation \(xy\frac{dy}{dx} = (x+2)(y+2)\), find the solution curve passing through the point \((1,-1)\).
\(y - x + 2 = \log\left(x^2(y+2)^2\right)\)
Find the equation of a curve passing through the point \((0,-2)\) given that at any point \((x,y)\) on the curve, the product of the slope of its tangent and y coordinate of the point is equal to the x coordinate of the point.
\(y^2 - x^2 = 4\)
At any point \((x,y)\) of a curve, the slope of the tangent is twice the slope of the line segment joining the point of contact to the point \((-4,-3)\). Find the equation of the curve given that it passes through \((-2,1)\).
\((x+4)^2 = y + 3\)
The volume of a spherical balloon being inflated changes at a constant rate. If initially its radius is 3 units and after 3 seconds it is 6 units, find the radius of the balloon after \(t\) seconds.
\((63t + 27)^{\frac{1}{3}}\)
In a bank, principal increases continuously at the rate of \(r\%\) per year. Find the value of \(r\) if Rs 100 doubles itself in 10 years \((\log_e 2 = 0.6931)\).
\(6.93\%\)
In a bank, principal increases continuously at the rate of 5% per year. An amount of Rs 1000 is deposited with this bank, how much will it be worth after 10 years \((e^{0.5} = 1.648)\).
Rs 1648
In a culture, the bacteria count is 1,00,000. The number is increased by 10% in 2 hours. In how many hours will the count reach 2,00,000, if the rate of growth of bacteria is proportional to the number present?
\(\frac{2\log 2}{\log\left(\frac{11}{10}\right)}\)
The general solution of the differential equation \(\frac{dy}{dx} = e^{x+y}\) is
\(e^x + e^{-y} = C\)
\(e^x + e^y = C\)
\(e^{-x} + e^y = C\)
\(e^{-x} + e^{-y} = C\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\((x^2+xy)\,dy=(x^2+y^2)\,dx\)
\((x-y)^2=Cx\,e^{-\frac{y}{x}}\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(y'=\frac{x+y}{x}\)
\(y=x\log|x|+Cx\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\((x-y)\,dy-(x+y)\,dx=0\)
\(\tan^{-1}\left(\frac{y}{x}\right)=-\frac{1}{2}\log(x^2+y^2)+C\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\((x^2-y^2)\,dx+2xy\,dy=0\)
\(x^2+y^2=Cx\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(x^2\frac{dy}{dx}=x^2-2y^2+xy\)
\(\frac{1}{2\sqrt{2}}\log\left|\frac{x+\sqrt{2}y}{x-\sqrt{2}y}\right|=\log|x|+C\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(x\,dy-y\,dx=\sqrt{x^2+y^2}\,dx\)
\(y+\sqrt{x^2+y^2}=Cx^2\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(\{x\cos(\frac{y}{x})+y\sin(\frac{y}{x})\}\,y\,dx=\{y\sin(\frac{y}{x})-x\cos(\frac{y}{x})\}\,x\,dy\)
\(xy\cos\left|\frac{y}{x}\right|=C\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(x\frac{dy}{dx}-y+x\sin(\frac{y}{x})=0\)
\(x\left[1-\cos\left(\frac{y}{x}\right)\right]=C\sin\left(\frac{y}{x}\right)\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\(y\,dx+x\log\left(\frac{y}{x}\right)\,dy-2x\,dy=0\)
\(cy=\log\left|\frac{y}{x}\right|-1\)
In each of the Exercises 1 to 10, show that the given differential equation is homogeneous and solve it:
\((1+e^{\frac{x}{y}})\,dx+e^{\frac{x}{y}}\left(1-\frac{x}{y}\right)\,dy=0\)
\(ye^{\frac{x}{y}}+x=C\)
For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
\((x+y)\,dy+(x-y)\,dx=0;\ y=1\ \text{when}\ x=1\)
\(\log(x^2+y^2)+2\tan^{-1}\left(\frac{y}{x}\right)=\frac{\pi}{2}+\log 2\)
For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
\(x^2\,dy+(xy+y^2)\,dx=0;\ y=1\ \text{when}\ x=1\)
\(y+2x=3x^2y\)
For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
\([x\sin^2(\frac{y}{x})-y]dx+x\,dy=0;\ y=\frac{\pi}{4}\ \text{when}\ x=1\)
\(\cot\left(\frac{y}{x}\right)=\log|ex|\)
For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
\(\frac{dy}{dx}-\frac{y}{x}+\cosec(\frac{y}{x})=0;\ y=0\ \text{when}\ x=1\)
\(\cos\left(\frac{y}{x}\right)=\log|ex|\)
For each of the differential equations in Exercises from 11 to 15, find the particular solution satisfying the given condition:
\(2xy+y^2-2x^2\frac{dy}{dx}=0;\ y=2\ \text{when}\ x=1\)
\(y=\frac{2x}{1-\log|x|}\ \ (x\ne 0,\ x\ne e)\)
A homogeneous differential equation of the form \(\frac{dx}{dy}=h\left(\frac{x}{y}\right)\) can be solved by making the substitution.
\(y=vx\)
\(v=yx\)
\(x=vy\)
\(x=v\)
Which of the following is a homogeneous differential equation?
\((4x+6y+5)\,dy-(3y+2x+4)\,dx=0\)
\((xy)\,dx-(x^3+y^3)\,dy=0\)
\((x^3+2y^2)\,dx+2xy\,dy=0\)
\(y^2\,dx+(x^2-xy-y^2)\,dy=0\)
For the differential equation, find the general solution:
\(\frac{dy}{dx}+2y=\sin x\)
\(y=\frac{1}{5}(2\sin x-\cos x)+Ce^{-2x}\)
For the differential equation, find the general solution:
\(\frac{dy}{dx}+3y=e^{-2x}\)
\(y=e^{-2x}+Ce^{-3x}\)
For the differential equation, find the general solution:
\(\frac{dy}{dx}+\frac{y}{x}=x^2\)
\(xy=\frac{x^4}{4}+C\)
For the differential equation, find the general solution:
\(\frac{dy}{dx}+(\sec x)y=\tan x\) \((0\le x<\frac{\pi}{2})\)
\(y(\sec x+\tan x)=\sec x+\tan x-x+C\)
For the differential equation, find the general solution:
\(\cos^2 x\,\frac{dy}{dx}+y=\tan x\) \((0\le x<\frac{\pi}{2})\)
\(y=(\tan x-1)+Ce^{-\tan x}\)
For the differential equation, find the general solution:
\(x\frac{dy}{dx}+2y=x^2\log x\)
\(y=\frac{x^2}{16}(4\log|x|-1)+Cx^{-2}\)
For the differential equation, find the general solution:
\(x\log x\,\frac{dy}{dx}+y=\frac{2}{x}\log x\)
\(y\log x=-\frac{2}{x}(1+\log|x|)+C\)
For the differential equation, find the general solution:
\((1+x^2)\,dy+2xy\,dx=\cot x\,dx\) \((x\ne 0)\)
\(y=(1+x^2)^{-1}\log|\sin x|+C(1+x^2)^{-1}\)
For the differential equation, find the general solution:
\(x\frac{dy}{dx}+y-x+xy\cot x=0\) \((x\ne 0)\)
\(y=\frac{1}{x}-\cot x+\frac{C}{x\sin x}\)
For the differential equation, find the general solution:
\((x+y)\frac{dy}{dx}=1\)
\(x+y+1=Ce^{y}\)
For the differential equation, find the general solution:
\(y\,dx+(x-y^2)\,dy=0\)
\(x=\frac{y^2}{3}+\frac{C}{y}\)
For the differential equation, find the general solution:
\((x+3y^2)\frac{dy}{dx}=y\) \((y>0)\)
\(x=3y^2+Cy\)
Find a particular solution satisfying the given condition:
\(\frac{dy}{dx}+2y\tan x=\sin x;\ y=0\ \text{when}\ x=\frac{\pi}{3}\)
\(y=\cos x-2\cos^2 x\)
Find a particular solution satisfying the given condition:
\((1+x^2)\frac{dy}{dx}+2xy=\frac{1}{1+x^2};\ y=0\ \text{when}\ x=1\)
\(y(1+x^2)=\tan^{-1}x-\frac{\pi}{4}\)
Find a particular solution satisfying the given condition:
\(\frac{dy}{dx}-3y\cot x=\sin 2x;\ y=2\ \text{when}\ x=\frac{\pi}{2}\)
\(y=4\sin^3 x-2\sin^2 x\)
Find the equation of a curve passing through the origin given that the slope of the tangent to the curve at any point \((x,y)\) is equal to the sum of the coordinates of the point.
\(x+y+1=e^x\)
Find the equation of a curve passing through the point \((0,2)\) given that the sum of the coordinates of any point on the curve exceeds the magnitude of the slope of the tangent to the curve at that point by 5.
\(y=4-x-2e^x\)
The Integrating Factor of the differential equation \(x\frac{dy}{dx}-y=2x^2\) is
\(e^{-x}\)
\(e^{-y}\)
\(\frac{1}{x}\)
\(x\)
The Integrating Factor of the differential equation \((1-y^2)\frac{dx}{dy}+yx=ay\) \((-1<y<1)\) is
\(\frac{1}{y^2-1}\)
\(\frac{1}{\sqrt{y^2-1}}\)
\(\frac{1}{1-y^2}\)
\(\frac{1}{\sqrt{1-y^2}}\)
For each of the differential equations given below, indicate its order and degree (if defined).
(i) \(\frac{d^2y}{dx^2}+5x\left(\frac{dy}{dx}\right)^2-6y=\log x\)
(ii) \(\left(\frac{dy}{dx}\right)^3-4\left(\frac{dy}{dx}\right)^2+7y=\sin x\)
(iii) \(\frac{d^4y}{dx^4}-\sin\left(\frac{d^3y}{dx^3}\right)=0\)
(i) Order 2; Degree 1
(ii) Order 1; Degree 3
(iii) Order 4; Degree not defined
For each of the exercises given below, verify that the given function (implicit or explicit) is a solution of the corresponding differential equation.
(i) \(xy=ae^x+be^{-x}+x^2\) : \(x\frac{d^2y}{dx^2}+2\frac{dy}{dx}-xy+x^2-2=0\)
(ii) \(y=e^x(a\cos x+b\sin x)\) : \(\frac{d^2y}{dx^2}-2\frac{dy}{dx}+2y=0\)
(iii) \(y=x\sin 3x\) : \(\frac{d^2y}{dx^2}+9y-6\cos 3x=0\)
(iv) \(x^2=2y^2\log y\) : \((x^2+y^2)\frac{dy}{dx}-xy=0\)
Prove that \(x^2-y^2=c(x^2+y^2)^2\) is the general solution of differential equation \((x^3-3xy^2)\,dx=(y^3-3x^2y)\,dy\), where \(c\) is a parameter.
Find the general solution of the differential equation \(\frac{dy}{dx}+\sqrt{\frac{1-y^2}{1-x^2}}=0\).
\(\sin^{-1}y+\sin^{-1}x=C\)
Show that the general solution of the differential equation \(\frac{dy}{dx}+\frac{y^2+y+1}{x^2+x+1}=0\) is given by \((x+y+1)=A(1-x-y-2xy)\), where \(A\) is parameter.
Find the equation of the curve passing through the point \((0,\frac{\pi}{4})\) whose differential equation is \(\sin x\cos y\,dx+\cos x\sin y\,dy=0\).
\(\cos y=\frac{\sec x}{\sqrt{2}}\)
Find the particular solution of the differential equation \((1+e^{2x})\,dy+(1+y^2)e^x\,dx=0\), given that \(y=1\) when \(x=0\).
\(\tan^{-1}y+\tan^{-1}(e^x)=\frac{\pi}{2}\)
Solve the differential equation \(y e^{\frac{x}{y}}\,dx=(x e^{\frac{x}{y}}+y^2)\,dy\) \((y\ne 0)\).
\(e^{\frac{x}{y}}=y+C\)
Find a particular solution of the differential equation \((x-y)(dx+dy)=dx-dy\), given that \(y=-1\), when \(x=0\). (Hint: put \(x-y=t\))
\(\log|x-y|=x+y+1\)
Solve the differential equation \(\left[\frac{e^{-2\sqrt{x}}}{\sqrt{x}}-\frac{y}{\sqrt{x}}\right]\frac{dx}{dy}=1\) \((x\ne 0)\).
\(y e^{2\sqrt{x}}=(2\sqrt{x}+C)\)
Find a particular solution of the differential equation \(\frac{dy}{dx}+y\cot x=4x\cosec x\) \((x\ne 0)\), given that \(y=0\) when \(x=\frac{\pi}{2}\).
\(y\sin x=2x^2-\frac{\pi^2}{2}\) \((\sin x\ne 0)\)
Find a particular solution of the differential equation \((x+1)\frac{dy}{dx}=2e^{-y}-1\), given that \(y=0\) when \(x=0\).
\(y=\log\left|\frac{2x+1}{x+1}\right|,\ x\ne -1\)
The general solution of the differential equation \(\frac{y\,dx-x\,dy}{y}=0\) is
\(xy=C\)
\(x=Cy^2\)
\(y=Cx\)
\(y=Cx^2\)
The general solution of a differential equation of the type \(\frac{dx}{dy}+P_1x=Q_1\) is
\(y\,e^{\int P_1\,dy}=\int\left(Q_1e^{\int P_1\,dy}\right)dy+C\)
\(y\,e^{\int P_1\,dx}=\int\left(Q_1e^{\int P_1\,dx}\right)dx+C\)
\(x\,e^{\int P_1\,dy}=\int\left(Q_1e^{\int P_1\,dy}\right)dy+C\)
\(x\,e^{\int P_1\,dx}=\int\left(Q_1e^{\int P_1\,dx}\right)dx+C\)
The general solution of the differential equation \(e^x\,dy+(ye^x+2x)\,dx=0\) is
\(xe^y+x^2=C\)
\(xe^y+y^2=C\)
\(ye^x+x^2=C\)
\(ye^y+x^2=C\)