Differential Equation and Applications (p-1) - Maharashtra Board STD 12 Commerce / Arts Maths (commerce) Questions

Q 1MCQ1 Mark

The integrating factor of $\frac{d y}{d x}- y = e ^{ x }$ is

A
$x$
B
$-x$
✓
$e^x$
D
$e^{-x}$

Answer: C.

Q 2MCQ1 Mark

Bacterial increases at a rate proportional to the number present. If the original number $M$ doubles in 3 hours, then the number of bacteria will be $4 M$ in

A
4 hours
✓
6 hours
C
8 hours
D
10 hours

Answer: B.

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Q 3MCQ1 Mark

The solution of $x \frac{d y}{d x}= y \log y$ is

A
$y=a e^x$
B
$y=b e^{2 x}$
C
$y=b e^{-2 x}$
✓
$y=e^{a x}$

Answer: D.

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Q 4MCQ1 Mark

The solution of $\frac{d y}{d x}+\frac{x^2}{y^2}=0$ is

A
$x^3+y^3=7$
B
$x^2+y^2=c$
✓
$x^3+y^3=c$
D
$x+y=c$

Answer: C.

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Q 5MCQ1 Mark

The solution of $\frac{d y}{d x}=1$ is

A
$x+y=c$
B
$x y=c$
C
$x^2+y^2=c$
✓
$y-x=c$

Answer: D.

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Q 6Write true or false if statement is wrong write the true statement.1 Mark

The degree of the differential equation $e^{\frac{d y}{d x}}=\frac{d y}{d x}+c$ is not defined.

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Q 7Write true or false if statement is wrong write the true statement.1 Mark

The power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any is called the order of the differential equation.

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Q 8Write true or false if statement is wrong write the true statement.1 Mark

The order of highest derivative occurring in the differential equation is called the degree of the differential equation.

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Q 9Write true or false if statement is wrong write the true statement.1 Mark

The degree of a differential equation is the power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any.

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Q 10Write true or false if statement is wrong write the true statement.1 Mark

The order and degree of a differential equation are always positive integers.

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Q 11Fil in The blanks.1 Mark

The differential equation by eliminating arbitrary constants from bx $+a y=a b$ is _________

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Q 12Fil in The blanks.1 Mark

The integrating factor of the differential equation $\frac{d y}{d x}- y = x$ is________

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Q 13Fil in The blanks.1 Mark

Order and degree of a differential equation are always _________ integers.

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Q 14Fil in The blanks.1 Mark

A solution of differential equation that can be obtained from the general solution by giving particular values to the arbitrary constants is called _________ solution.

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Q 15Fil in The blanks.1 Mark

The power of the highest ordered derivative when all the derivatives are made free from negative and/or fractional indices if any is called ________ of the differential equation.

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Q 16Solve the following Question.(1 Marks)1 Mark

Solve the following differential equations:$dr =\operatorname{ar} d \theta-\theta dr$

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Q 17Solve the following Question.(1 Marks)1 Mark

Solve the following differential equations:$e^{d y / d x}=x$

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Q 18Solve the following Question.(1 Marks)1 Mark

Solve the following differential equations:$\frac{d y}{d x}=1+ x + y + xy$

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Q 19Solve the following Question.(1 Marks)1 Mark

Find the order and degree of the following differential equations:
$x+\frac{d y}{d x}=1+\left(\frac{d y}{d x}\right)^2$

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Q 20Solve the following Question.(1 Marks)1 Mark

Find the order and degree of the following differential equations:
$\left[\frac{d^3 y}{d x^3}+x\right]^{3 / 2}=\frac{d^2 y}{d x^2}$

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Q 21Solve the Following Question.(2 Marks)2 Marks

Verify that $y=\log x+c$ is a solution of the differential equation
$x \frac{d^2 y}{d x^2}+\frac{d y}{d x}=0$

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Q 22Solve the Following Question.(2 Marks)2 Marks

Solve $\frac{d y}{d x}=\log xdx$

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Q 23Solve the Following Question.(2 Marks)2 Marks

Solve the following differential equations : $d y+(2 y) d x=8 d x$

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Q 24Solve the Following Question.(2 Marks)2 Marks

Solve the following differential equations : $\frac{d y}{d x}+ y =3$

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Q 25Solve the Following Question.(2 Marks)2 Marks

Solve the following differential equations : $\frac{d y}{d x}+y=e^{-x}$

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Q 26Solve the Following Question.(3 Marks)3 Marks

Solve $\frac{d y}{d x}+\frac{2}{x} y=x^2$

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Q 27Solve the Following Question.(3 Marks)3 Marks

Solve $\left( x +2 y ^3\right) \frac{d y}{d x}= y$

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Q 28Solve the Following Question.(3 Marks)3 Marks

The resale value of a machine decreases over a 10 years period at a rate that depends on the age of the machine. When the machine is $x$ years old, the rate at which its value is changing is $₹ 2200(x-10$ ) per year. Express the value of the machine as a function of its age and initial value. If the machine was originally worth ₹ $1,20,000$ how much will it be worth when it is 10 years old?

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Q 29Solve the Following Question.(3 Marks)3 Marks

Solve the following differential equations : $(x+a) \frac{d y}{d x}=- y + a$

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Q 30Solve the Following Question.(3 Marks)3 Marks

Solve the following differential equations : $\frac{d y}{d x}+2 x y=x$

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Q 31Solve the Following Question.(4 Marks)4 Marks

Solve $(x+y) d y=a^2 d x$

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Q 32Solve the Following Question.(4 Marks)4 Marks

Solve $\frac{d y}{d x}=\frac{x+y+1}{x+y-1}$ when $x =\frac{2}{3}$ and $y =\frac{1}{3}$.

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Q 33Solve the Following Question.(4 Marks)4 Marks

Solve the following differential equations:Find the differential equation of the family of curves $y=e^x(a x+$ $b x^2$ ), where $a$ and $b$ are arbitrary constants.

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Q 34Solve the Following Question.(4 Marks)4 Marks

Solve $yx \frac{d y}{d x}= x ^2+2 y ^2$

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Q 35Solve the Following Question.(4 Marks)4 Marks

For each of the following differential equations find the particular solution : $\frac{d y}{d x}=4 x + y +1$, when $y =1, x =0$

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Q 36Solve the Following Question.(5 Marks)5 Marks

The rate of growth of the population is proportional to the number present. If the population doubled in the last 25 years and the present population is 1 lakh, when will the city have a population of 400000 ?

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Q 37Solve the Following Question.(5 Marks)5 Marks

Solve y $\log y \frac{d x}{d y}+ x -\log y =0$

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Q 38Solve the Following Question.(5 Marks)5 Marks

Solve $y d x-x d y=-\log x d x$

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Q 39Solve the Following Question.(5 Marks)5 Marks

$y \log y \frac{d x}{d y}=\log y - x$

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Q 40Solve the Following Question.(5 Marks)5 Marks

Solve $y d x-x d y+\log x d x=0$