# Differential Equations Worksheet

Differential Equations Worksheet
• Page 1
1.
Which of the following is the particular solution for the given differential equation such that $y$ = 3 when $x$ = 0?
 a. $y$2 = ($\frac{2}{3}$)$x$3 + 9 b. $y$2 = -$\frac{2}{3x³}$ + 9 c. $y$2 = $x$3 + 9 d. $y$2 = $\frac{2}{3}x³$ - 18

#### Solution:

dydx =x²y
[Given.]

ydy = x2dx

ydy = x2dx
[Integrate both sides.]

y²2 =x³3 + C

9 / 2= C
[Substitute x = 0 and y = 3.]

So, the required particular solution is y2 = 2x³3 + 9.

2.
Which of the following is the particular solution for the given differential equation given that $y$ = 4 when $x$ = 0?
 a. - 4 b. - 4 c. - 4 d. - 4

#### Solution:

dydx =2x + 1y - 3
[Given.]

(y - 3)dy = (2x + 1) dx

(y - 3)dy = (2x + 1)dx
[Integrate both sides.]

y²2 - 3y = x² + x + C

(16 / 2) - 12 = C
[Substitute x = 0 and y = 4.]

C = - 4
[Simplify.]

The particular solution is at x = 0, y = 4 is y²2 - 3y = x² + x - 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 + $x$ #### Solution: C′(x) = 2x + 3x2 [Given.] Since, cost of 2 units is$12, we have C(2) = 12.
[Given.]

C(x) = (2x + 3x2)dx
[Integrate both sides.]

C(x) = x2 + x3 + k

12 = 22 + 23 + C
[Substitute x = 2, C(2) = 12.]

12 = 4 + 8 + C

C = 0

Thus, the cost function is C(x) = x2 + x3.

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

#### Solution:

Any function that satisfies a differential equation is known as a particular solution of that differential equation.

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

#### Solution:

The equation that involves one (or) more derivatives of some unknown functions which are required to be found is called a Differential equation.
[Definition.]

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

#### Solution:

The order of the highest order derivative of the unknown function occurring in the differential equation is known as the Order of that differential equation.
[Definition.]

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

#### Solution:

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.
[Definition.]

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

#### Solution:

The highest order derivative in the differential equation 5(y″)5 + 10y′ + 40y = 0 is y″ and hence the order of the differential equation is k = 2
[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.]

k2 + 5k = (2)2 + (5)(2) = 14
[Substitute k = 2 to find the value of k2 + 5k.]

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

#### Solution:

The highest order derivative in the differential equation 43(y′)2 - 8 y′ - 6 y= 0 is y′, whose highest power is 2. So, the degree of the differential equation is a = 2
[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.]

4a2 + a + 6 = 4(2)2 + 2 + 6 = 24
[Substitute the value of a to find the value of 4a2 + a + 6.]

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

#### Solution:

The highest order derivative in the differential equation 19(y″′)8 + 17(y″)7 + 8(y′)5 + 10 y = 0 is y″′, whose order is 3. So, the order of the differential equation is 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.]