Worksheet: Indefinite Integrals and Initial Value Problems

In this worksheet, we will practice using integration to find particular solutions to initial value problems involving differential equations of the form y' = f(x).

Q1:

Find a particular solution for the following differential equation for which 𝑦(0)=12: dd𝑦𝑥=8𝑥+3.

  • A 𝑦 = 8 𝑥 + 3 𝑥
  • B 𝑦 = 4 𝑥 + 3 𝑥
  • C 𝑦 = 4 𝑥 + 3 𝑥 + 1 2
  • D 𝑦 = 8 𝑥 + 3 𝑥 + 1 2

Q2:

Find the solution for the following differential equation for 𝑦(0)=0: dd𝑦=𝑒𝑥.

  • A 𝑒 + 𝑒 = 2
  • B 𝑒 + 𝑒 = 2
  • C 𝑒 + 𝑒 = 2
  • D 𝑒 + 𝑒 = 2

Q3:

Find the solution for the following differential equation for 𝑦(1)=1: dd𝑦𝑥+𝑦=0.

  • A 𝑦 = 𝑒
  • B 𝑦 = 𝑒
  • C 𝑦 = 𝑒
  • D 𝑦 = 𝑒

Q4:

Find the solution for the following differential equation for 𝑦(0)=(2)ln: 𝑒𝑦𝑥2𝑥=0.dd

  • A 𝑦 = | | 𝑥 + 2 | | l n
  • B 𝑦 = | | 𝑥 2 + 2 | | l n
  • C 𝑦 = | 𝑥 + 2 | l n
  • D 𝑦 = | 2 𝑥 + 2 | l n

Q5:

Find the particular solution for the following separable differential equation: cosdd(𝑦)𝑦𝑥𝑥=0,𝑦(0)=0.

  • A 𝑦 = 𝑥 2
  • B 𝑦 = 𝑥 2 s i n
  • C 𝑦 = 𝑥 2 c o s
  • D 𝑦 = 𝑥 2 s i n

Q6:

Find the solution for the following differential equation for 𝑦(0)=16: dd𝑦𝑥=4𝑦20.

  • A 𝑦 = 5 + 1 1 𝑒
  • B 𝑦 = 5 + 1 1 𝑒
  • C 𝑦 = 5 + 1 1 𝑒
  • D 𝑦 = 5 + 1 1 𝑒

Q7:

Find the solution for the following differential equation for 𝑦(0)=2: 1𝑒𝑦𝑥=0,𝑦(0)=2.dd

  • A 𝑦 = ( 2 𝑒 + 2 )
  • B 𝑦 = ( 2 𝑒 + 2 )
  • C 𝑦 = 1 + 𝑒
  • D 𝑦 = ( 2 𝑒 + 2 )

Q8:

Find the solution for the following differential equation for 𝑦(0)=1: dd𝑦𝑥𝑥𝑥=0.

  • A 𝑦 = 𝑥 2 + 𝑥 3 + 1
  • B 𝑦 = 𝑥 2 𝑥 3 + 1
  • C 𝑦 = 2 𝑥 3 𝑥 + 1
  • D 𝑦 = 2 𝑥 + 3 𝑥 + 1

Q9:

For a circuit containing a capacitor and a resistor, the first-order differential equation that describes a discharging capacitor is 𝑄𝐶+𝑅𝑄𝑡=0.dd If 𝑄 represents the charge within the capacitor at 𝑡=0seconds, find the general solution. 𝐶 is the capacitance of the capacitor. 𝑅 is the resistance of the resistor.

  • A 𝑄 = 𝑄 𝑒
  • B 𝑄 = 𝑄 𝑒
  • C 𝑄 = 𝑄 𝑒
  • D 𝑄 = 𝑄 𝑒

Q10:

Find the solution of the differential equation ddsec𝑢𝑡=𝑡+𝑡𝑢 that satisfies the initial condition 𝑢(0)=3.

  • A 𝑢 = 𝑡 + 2 𝑡 + 9 t a n
  • B 𝑢 = 𝑡 + 2 𝑡 9 t a n
  • C 𝑢 = 𝑡 2 𝑡 + 9 t a n
  • D 𝑢 = 𝑡 + 2 𝑡 t a n
  • E 𝑢 = 𝑡 + 𝑡 + 9 t a n

Q11:

Find the solution of the differential equation 𝑦𝑥=𝑎+𝑦tan, where 0<𝑥<𝜋2, that satisfies the initial condition 𝑦𝜋3=𝑎.

  • A 𝑦 = 4 𝑎 𝑥 𝑎 s i n
  • B 𝑦 = 4 𝑎 𝑥 𝑎 c o s
  • C 𝑦 = 𝑎 4 𝑎 3 𝑥 s i n
  • D 𝑦 = 4 𝑎 3 𝑥 𝑎 s i n
  • E 𝑦 = 𝑎 4 𝑎 𝑥 c o s

Q12:

Solve the differential equation dd𝑦𝑥𝑥9=1 for 𝑦 given that 𝑦(5)=3ln.

  • A 𝑦 = 𝑥 + 𝑥 9 + 3 l n l n
  • B 𝑦 = 𝑥 + 𝑥 9 3 l n l n
  • C 𝑦 = 1 3 𝑥 9 𝑥 + 1 3 4 5 l n l n
  • D 𝑦 = 1 3 𝑥 9 𝑥 + 1 3 4 5 l n l n
  • E 𝑦 = 𝑥 + 𝑥 9 2 3 l n l n

Q13:

Find the solution for the following differential equation for 𝑦(0)=0: (1𝑥)𝑦=𝑥(𝑦+1)𝑥.dd

  • A ( 𝑦 + 1 ) ( 1 𝑥 ) = 𝑒
  • B ( 𝑦 1 ) ( 1 + 𝑥 ) = 𝑒
  • C ( 𝑦 + 1 ) ( 1 𝑥 ) = 𝑒
  • D ( 𝑦 1 ) ( 1 + 𝑥 ) = 𝑒

Q14:

You are told that 𝑓(𝑥)=12[𝑒+𝑒]. If 𝑓(0)=1 and 𝑓(0)=0, which of the following is equal to 𝑓(𝑥)?

  • A 𝑓 ( 𝑥 )
  • B 𝑓 ( 𝑥 )
  • C 𝑓 ( 𝑥 )
  • D 𝑓 ( 𝑥 )

Q15:

Find the function whose first derivative is 27𝑥83𝑥2 given the value of the function equals 1 when 𝑥 equals 1.

  • A 𝑓 ( 𝑥 ) = 3 𝑥 + 3 𝑥 + 4 𝑥 + 3
  • B 𝑓 ( 𝑥 ) = 9 𝑥 + 6 𝑥 + 4 𝑥 + 6
  • C 𝑓 ( 𝑥 ) = 2 7 𝑥 4 8 𝑥 6 3 4
  • D 𝑓 ( 𝑥 ) = 3 𝑥 2 𝑥 6

Q16:

Suppose that ddcsc𝑦𝑥=8𝑥 and 𝑦=12 when 𝑥=𝜋3. Find 𝑦 in terms of 𝑥.

  • A 8 𝑥 + 1 2 c o t
  • B 8 𝑥 8 3 3 + 1 2 c o t
  • C 8 𝑥 1 2 + 8 3 3 c o t
  • D 8 𝑥 1 2 + 8 3 3 c o t
  • E 8 𝑥 1 2 c o t

Q17:

Find the function𝑓, if 𝑓(𝑡)=2𝑡(𝑡+4𝑡)sectansec, when 𝜋2<𝑡<𝜋2 and 𝑓𝜋3=2.

  • A 𝑓 ( 𝑡 ) = 8 𝑡 + 2 𝑡 6 + 8 3 t a n s e c
  • B 𝑓 ( 𝑡 ) = 8 𝑡 + 2 𝑡 8 3 + 2 t a n s e c
  • C 𝑓 ( 𝑡 ) = 4 𝑡 + 𝑡 6 + 8 3 t a n s e c
  • D 𝑓 ( 𝑡 ) = 2 𝑡 + 8 𝑡 8 3 + 2 t a n s e c
  • E 𝑓 ( 𝑡 ) = 2 𝑡 + 8 𝑡 6 + 8 3 t a n s e c

Q18:

Determine the function 𝑓 satisfying 𝑓(𝜃)=2𝜃+3𝜃sincos, 𝑓(0)=5, and 𝑓(0)=1.

  • A 𝑓 ( 𝜃 ) = 3 𝜃 2 𝜃 + 3 s i n c o s
  • B 𝑓 ( 𝜃 ) = 2 𝜃 3 𝜃 1 s i n c o s
  • C 𝑓 ( 𝜃 ) = 3 𝜃 2 𝜃 3 𝜃 2 s i n c o s
  • D 𝑓 ( 𝜃 ) = 3 𝜃 2 𝜃 3 𝜃 + 8 s i n c o s
  • E 𝑓 ( 𝜃 ) = 𝜃 2 𝜃 3 𝜃 + 2 s i n c o s

Q19:

Determine the function 𝑓 if 𝑓(𝑥)=4𝑥cos, 𝑓(0)=1, 𝑓(0)=4, and 𝑓(0)=4.

  • A 𝑓 ( 𝑥 ) = 4 𝑥 + 8 𝑥 4 𝑥 1 s i n
  • B 𝑓 ( 𝑥 ) = 2 𝑥 + 8 𝑥 + 𝑥 + 1 s i n
  • C 𝑓 ( 𝑥 ) = 4 𝑥 8 𝑥 4 𝑥 1 s i n
  • D 𝑓 ( 𝑥 ) = 2 𝑥 + 8 𝑥 4 𝑥 1 s i n
  • E 𝑓 ( 𝑥 ) = 4 𝑥 8 𝑥 4 𝑥 + 1 s i n

Q20:

Suppose that ddsincos𝑦𝑥=92𝑥35𝑥 and 𝑦=7 when 𝑥=𝜋6. Find 𝑦 in terms of 𝑥.

  • A 𝑦 = 3 5 5 𝑥 + 9 2 2 𝑥 + 1 0 1 2 0 s i n c o s
  • B 𝑦 = 9 2 𝑥 3 5 𝑥 + 1 3 s i n c o s
  • C 𝑦 = 9 2 5 𝑥 + 3 5 2 𝑥 + 8 9 2 0 s i n c o s
  • D 𝑦 = 3 5 2 𝑥 9 2 5 𝑥 + 8 9 2 0 s i n c o s
  • E 𝑦 = 9 2 2 𝑥 3 5 5 𝑥 + 8 9 2 0 s i n c o s

Q21:

Determine the function 𝑓 if 𝑓(𝑥)=𝑥+1, 𝑓(1)=5, and 𝑓(1)=3.

  • A 𝑓 ( 𝑥 ) = 𝑥 3 0 + 𝑥 2 1 1 𝑥 5 7 3
  • B 𝑓 ( 𝑥 ) = 𝑥 3 0 + 𝑥 2 + 1 1 𝑥 5 + 7 3
  • C 𝑓 ( 𝑥 ) = 𝑥 3 0 + 𝑥 2 + 2 1 𝑥 5 + 1 3
  • D 𝑓 ( 𝑥 ) = 𝑥 5 + 1 6 𝑥 5
  • E 𝑓 ( 𝑥 ) = 𝑥 5 + 2 6 𝑥 5 + 1 3

Q22:

Find the function 𝑓 on (0,) which satisfies 𝑓(1)=3, 𝑓(4)=0, and 𝑓(𝑥)=4𝑥.

  • A 𝑓 ( 𝑥 ) = 𝑥 1 + 4 3 4 4 𝑥 4 4 3 4 l n l n l n
  • B 𝑓 ( 𝑥 ) = 𝑥 1 + 4 3 4 4 𝑥 + 4 3 4 + 4 l n l n l n
  • C 𝑓 ( 𝑥 ) = 𝑥 1 + 4 3 4 4 𝑥 + 4 3 4 + 4 l n l n l n
  • D 𝑓 ( 𝑥 ) = 4 𝑥 3 4 4 𝑥 4 3 4 l n l n l n
  • E 𝑓 ( 𝑥 ) = 𝑥 1 + 4 3 4 4 𝑥 4 4 3 4 l n l n l n

Q23:

Determine the function 𝑓 satisfying 𝑓(𝑥)=12𝑥10𝑥+3, 𝑓(0)=5, and 𝑓(0)=2.

  • A 𝑓 ( 𝑥 ) = 4 𝑥 5 𝑥 + 3 𝑥 + 5
  • B 𝑓 ( 𝑥 ) = 4 𝑥 5 𝑥 + 3 𝑥 + 2
  • C 𝑓 ( 𝑥 ) = 𝑥 5 𝑥 3 + 3 𝑥 2 + 5 𝑥 + 2
  • D 𝑓 ( 𝑥 ) = 𝑥 5 𝑥 3 + 3 𝑥 2 + 2 𝑥 + 5
  • E 𝑓 ( 𝑥 ) = 1 2 𝑥 1 0 𝑥 + 3 𝑥 + 2 𝑥 + 5

Q24:

The function 𝑓(𝑥) satisfies the relation 𝑓(𝑎+)𝑓(𝑎)=2𝑎𝑘+2 where 𝑎, and 𝑘 is a constant. Find 𝑓(𝑥) given 𝑓(4)=8 and 𝑓(2)=4.

  • A 𝑥 8
  • B 𝑥 + 1 2
  • C 2 𝑥
  • D 𝑥

Q25:

Given that ddcos𝑠𝑡=99𝑡4, and 𝑠=4 when 𝑡=4𝜋3, find the relation between 𝑠 and 𝑡.

  • A 𝑠 ( 𝑡 ) = 9 9 𝑡 4 4 s i n
  • B 𝑠 ( 𝑡 ) = 9 9 𝑡 4 4 s i n
  • C 𝑠 ( 𝑡 ) = 4 9 𝑡 4 4 s i n
  • D 𝑠 ( 𝑡 ) = 4 9 𝑡 4 4 s i n

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.