HSC Arts (English Medium) १२ वीं कक्षा - Maharashtra State Board Question Bank Solutions

In the following example verify that the given function is a solution of the differential equation.

`"y" = "e"^"ax" sin "bx"; ("d"^2"y")/"dx"^2 - 2"a" "dy"/"dx" + ("a"^2 + "b"^2)"y" = 0`

In the following example verify that the given function is a solution of the differential equation.

`"y" = 3 "cos" (log "x") + 4 sin (log "x"); "x"^2 ("d"^2"y")/"dx"^2 + "x" "dy"/"dx" + "y" = 0`

In the following example verify that the given function is a solution of the differential equation.

`"xy" = "ae"^"x" + "be"^-"x" + "x"^2; "x" ("d"^2"y")/"dx"^2 + 2 "dy"/"dx" + "x"^2 = "xy" + 2`

In the following example verify that the given function is a solution of the differential equation.

`"x"^2 = "2y"^2 log "y",  "x"^2 + "y"^2 = "xy" "dx"/"dy"`

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = a sin (x + b)

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

(y - a)² = b(x + 4)

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = `sqrt("a" cos (log "x") + "b" sin (log "x"))`

Obtain the differential equation by eliminating the arbitrary constants from the following equation:

y = `"Ae"^(3"x" + 1) + "Be"^(- 3"x" + 1)`

Form the differential equation of all parabolas which have 4b as latus rectum and whose axis is parallel to the Y-axis.

Form the differential equation of all the lines which are normal to the line 3x + 2y + 7 = 0.

Form the differential equation of the hyperbola whose length of transverse and conjugate axes are half of that of the given hyperbola `"x"^2/16 - "y"^2/36 = "k"`.

Solve the following differential equation:

`"dy"/"dx" = "x"^2"y" + "y"`

Solve the following differential equation:

`"dy"/"dx" = ("2y" - "x")/("2y + x")`

Solve the following differential equation:

x dy = (x + y + 1) dx

Solve the following differential equation:

`"dy"/"dx" + "y cot x" = "x"^2 "cot x" + "2x"`

Solve the following differential equation:

y log y = (log y² - x) `"dy"/"dx"`

Solve the following differential equation:

`"dx"/"dy" + "8x" = 5"e"^(- 3"y")`

Find the particular solution of the following differential equation:

`("x + 2y"^2) "dy"/"dx" = "y",` when x = 2, y = 1

Find the particular solution of the following differential equation:

`"dy"/"dx" - 3"y" cot "x" = sin "2x"`, when `"y"(pi/2) = 2`

Find the particular solution of the following differential equation:

(x + y)dy + (x - y)dx = 0; when x = 1 = y