Find the differential equation of the family of straight lines y = mx + c when
(i) m is the arbitrary constant
(ii) c is the arbitrary constant
(iii) m and c both are arbitrary constants.

Solve 3e^xtan ydx +(1 + e^x)sec²ydy = 0 given y(0) = \(\frac { \pi }{ 4 } \)

Solve : x - y \(\frac { dx }{ dy } =a\left( { x }^{ 2 }+\frac { dx }{ dy } \right) \)

The normal lines to a given curve at each point(x,y) on the curve pass through the point (1, 0). The curve passes through the point (1, 2). Formulate the differential equation representing the problem and hence find the equation of the curve.

The sum of Rs. 2,000 is compounded continuously, the nominal rate of interest being 5% per annum. In how many years will the amount be double the original principal? (log_e2 = 0.6931)

Solve the differential equation y²dx + (xy + x²)dy = 0

Solve the differential equation \(\frac { dy }{ dx } =\frac { x-y }{ x+y } \)

Find the particular solution of the differential equation x² dy + y(x + y)dx = 0 given  that x = 1, y = 1

If the marginal cost of producing x shoes is given by (3xy + y²)dx + (x² + xy)dy = 0 and the total cost of producing a pair of shoes is given by Rs. 12. Then find the total cost  function.

The marginal revenue ‘y’ of output ‘q’ is given by the equation \(\frac { dy }{ dq } =\frac { { q }^{ 2 }+{ 3 }y^{ 2 } }{ 2qy } \). Find  the total Revenue function when output is 1 unit and Revenue is Rs. 5.

Solve the following homogeneous differential equations.
\(x\frac { dy }{ dx } =x+y\).

Solve the following homogeneous differential equations.
\((x-y)\frac { dy }{ dx } =x+3y\).

Solve the following homogeneous differential equations.
\(x\frac { dy }{ dx } -y=\sqrt { { x }^{ 2 }+{ y }^{ 2 } } \)

Solve the following homogeneous differential equations.
\(\frac { dy }{ dx } =\frac { 3x-2y }{ 2x-3y } \)

Solve the following homogeneous differential equations.
(y² − 2xy)dx = (x² − 2xy)dy

The slope of the tangent to a curve at any point (x, y) on it is given by (y³−2yx²)dx + (2xy²−x³)dy = 0 and the curve passes through (1, 2). Find the equation of the curve.

An electric manufacturing company makes small household switches. The company estimates the marginal revenue function for these switches to be (x² + y²)dy = xydx where x represents the number of units (in thousands). What is the total revenue function?

Solve cos² x \(\frac{dy}{dx}\) + y = tan x

Solve (x² + 1)\(\frac { dy }{ dx } \) + 2xy = 4x² 

Solve \(\frac { dy }{ dx } \) −3ycot x = sin 2x given that y = 2 when x = \(\frac { \pi }{ 2 } \)

A firm has found that the cost C of producing x tons of certain product by the equation x\(\frac { dC }{ dx } =\frac { 3 }{ x } -C\) and C = 2 when x = 1. Find the relationship between C and x.

Solve the following:
\(x\frac { dy }{ dx } +2y={ x }^{ 4 }\)

Solve the following:
\(\frac{d y}{d x}+\frac{3 x^{2}}{1+x^{3}} y=\frac{1+x^{2}}{1+x^{3}}\)

Solve the following:
\(\frac { dy }{ dx } +\frac { y }{ x } ={ xe }^{ x }\)

Solve the following:
\(\frac { dy }{ dx } \)+ ytan x = cos³ x

If \(\frac { dy }{ dx } \) + 2y tan x = sin x and if y = 0 when x = \(\frac{\pi}{3}\) express y in terms of x.

\(\frac { dy }{ dx } +\frac { y }{ x } ={ xe }^{ x }\)

A bank pays interest by continuous compounding, that is by treating the interest rate as the instantaneous rate of change of principal. A man invests Rs. 1,00,000 in the bank deposit which accures interest, 8% per year compounded continuously. How much will he get after 10 years.

(D² − 3D + 2)y = e^3x which shall vanish for x = 0 and for x = log 2

Solve the following differential equations (D²+D−6)y=e^3x + e^−3x

Solve the following differential equations (D²−10D+25)y = 4e^5x + 5

Solve the following differential equations (4D² + 16D + 15)y = \({ 4e }^{ -\frac { 3 }{ 2 } x }\)

Solve the following differential equations (3D² + D − 14)y = 13e^2x

Suppose that the quantity demanded Q₄ = 13 - 6P + 2\(\frac { dp }{ dt } +\frac { { d }^{ 2 }p }{ { dt }^{ 2 } } \) and quantity supplied Q_s = − 3 + 2p, where p is the price. Find the equilibrium price for market clearance.

Solve: (D² − 2D + 1)y = e^2x + e^x

Solve: (3D² + D - 14)y = 4 - 13\({ e }^{\frac{-7}{3}x}\)

Suppose that the quantity demanded \({ Q }_{ d }=29-2p-5\frac { dp }{ dt } +\frac { { d }^{ 2 }p }{ { dt }^{ 2 } } \) and quantity supplied Q_s = 5 + 4p where p is the price. Find the equilibrium price for market clearance.

Suppose that \({ Q }_{ d }=30-5P+2\frac { dp }{ dt } +\frac { { d }^{ 2 }P }{ { dt }^{ 2 } } \) and Q_s = 6 + 3P. Find the equilibrium price for market clearance.

Solve (x² + y²)dx + 2xy dy = 0

A manufacturing company has found that the cost C of operating and maintaining the equipment is related to the length ‘m’ of intervals between overhauls by the equation m²\(\frac{dC}{dm}\) + 2mC = 2 and c = 4 and when m = 2. Find the relationship between C and m.

Solve (D²- 3D + 2)y = e^4x given y = 0 when x = 0 and x = 1.

Solve \(\frac { dy }{ dx } +ycosx+x=2cosx\).

Solve x²ydx-(x³+y³)dy = 0