Differential Equations Worksheet

**Page 1**

1.

Which of the following is the particular solution for the given differential equation $\frac{dy}{dx}=\frac{x\xb2}{y}$ such that $y$ = 3 when $x$ = 0?

a. | $y$ ^{2} = ($\frac{2}{3}$)$x$^{3} + 9 | ||

b. | $y$ ^{2} = -$\frac{2}{3x\xb3}$ + 9 | ||

c. | $y$ ^{2} = $x$^{3} + 9 | ||

d. | $y$ ^{2} = $\frac{2}{3}x\xb3$ - 18 |

[Given.]

[Integrate both sides.]

[Substitute

So, the required particular solution is

Correct answer : (1)

2.

Which of the following is the particular solution for the given differential equation $\frac{dy}{dx}=\frac{2x+1}{y-3}$ given that $y$ = 4 when $x$ = 0?

a. | $\frac{y\xb2}{2}+3y=x\xb2+x$ - 4 | ||

b. | $\frac{y\xb2}{2}-3y=x\xb2-x$ - 4 | ||

c. | $\frac{y\xb2}{2}+3y=x\xb2-x$ - 4 | ||

d. | $\frac{y\xb2}{2}-3y=x\xb2+x$ - 4 |

[Given.]

(

[Integrate both sides.]

(

[Substitute

C = - 4

[Simplify.]

The particular solution is at

Correct answer : (4)

3.

Which of the following will be the cost function for the marginal cost function C′($x$) = 2$x$ + 3$x$^{2} when cost of 2 units is $12?

a. | C($x$) = - $x$ ^{2} + $x$^{3} | ||

b. | C($x$) = $x$ ^{2} - $x$^{3} | ||

c. | C($x$) = $x$ ^{2} + $x$^{3} | ||

d. | C($x$) = $x$ ^{2} + $x$^{3} + $x$ |

[Given.]

Since, cost of 2 units is $12, we have C(2) = 12.

[Given.]

C(

[Integrate both sides.]

C(

12 = 2

[Substitute

12 = 4 + 8 +

Thus, the cost function is C(

Correct answer : (3)

4.

Any function that satisfies a differential equation is known as the ____ of that differential equation.

a. | General solution | ||

b. | Order | ||

c. | Particular solution | ||

d. | Degree |

Correct answer : (3)

5.

The equation that involves one (or) more derivatives of some unknown functions which are required to be found is called

a. | Non linear equation | ||

b. | Differential equation | ||

c. | Linear equation | ||

d. | Quadratic equation |

[Definition.]

Correct answer : (2)

6.

The order of the highest order derivative of the unknown function occurring in the differential equation is the _______ of the differential equation.

a. | Order | ||

b. | Degree | ||

c. | Particular solution | ||

d. | None of the above |

[Definition.]

Correct answer : (1)

7.

The highest power of the highest-order derivative in a differential equation is ________ of the differential equation.

a. | Degree | ||

b. | Order | ||

c. | Power | ||

d. | Both A and B |

[Definition.]

Correct answer : (1)

8.

If the order of the differential equation 5($y$″)^{5} + 10$y$′ + 40$y$ = 0 is $k$, then find the value of $k$^{2} + 5$k$.

a. | 50 | ||

b. | 14 | ||

c. | 30 | ||

d. | 10 |

[The order of the highest order derivative of a differential equation whose differential coefficients have positive integral powers is the order of that differential equation.]

[Substitute

Correct answer : (2)

9.

If the degree of the differential equation 43($y$′)^{2} - 8 $y$′ - 6 $y$= 0 is $a$ then find the value of 4$a$^{2} + $a$ + 6.

a. | 11 | ||

b. | 18 | ||

c. | 16 | ||

d. | 24 |

[The degree of a differential equation whose differential coefficients have positive integral powers is defined as the highest power(positive integral index) of the highest order derivative in it.]

[Substitute the value of

Correct answer : (4)

10.

Write the order of the differential equation 19($y$″′)^{8} + 17($y$″)^{7} + 8($y$′)^{5} + 10 $y$ = 0.

a. | 1 | ||

b. | 7 | ||

c. | 8 | ||

d. | 3 |

[The order of the highest order derivative of a differential equation whose differential coefficients have positive integral powers is the order of that differential equation.]

Correct answer : (4)