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.

  • A𝑦=8𝑥+3𝑥
  • B𝑦=4𝑥+3𝑥
  • C𝑦=4𝑥+3𝑥+12
  • D𝑦=8𝑥+3𝑥+12

Q2:

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

  • A𝑒+𝑒=2
  • B𝑒+𝑒=2
  • C𝑒+𝑒=2
  • D𝑒+𝑒=2

Q3:

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

  • A𝑦=𝑒
  • B𝑦=𝑒
  • C𝑦=𝑒
  • D𝑦=𝑒

Q4:

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

  • A𝑦=||𝑥+2||ln
  • B𝑦=||𝑥2+2||ln
  • C𝑦=|𝑥+2|ln
  • D𝑦=|2𝑥+2|ln

Q5:

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

  • A𝑦=𝑥2
  • B𝑦=𝑥2sin
  • C𝑦=𝑥2cos
  • D𝑦=𝑥2sin

Q6:

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

  • A𝑦=5+11𝑒
  • B𝑦=5+11𝑒
  • C𝑦=5+11𝑒
  • D𝑦=5+11𝑒

Q7:

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

  • A𝑦=(2𝑒+2)
  • B𝑦=(2𝑒+2)
  • C𝑦=1+𝑒
  • D𝑦=(2𝑒+2)

Q8:

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

  • A𝑦=𝑥2+𝑥3+1
  • B𝑦=𝑥2𝑥3+1
  • C𝑦=2𝑥3𝑥+1
  • D𝑦=2𝑥+3𝑥+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.

  • A𝑄=𝑄𝑒
  • B𝑄=𝑄𝑒
  • C𝑄=𝑄𝑒
  • D𝑄=𝑄𝑒

Q10:

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

  • A𝑢=𝑡+2𝑡+9tan
  • B𝑢=𝑡+2𝑡9tan
  • C𝑢=𝑡2𝑡+9tan
  • D𝑢=𝑡+2𝑡tan
  • E𝑢=𝑡+𝑡+9tan

Q11:

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

  • A𝑦=4𝑎𝑥𝑎sin
  • B𝑦=4𝑎𝑥𝑎cos
  • C𝑦=𝑎4𝑎3𝑥sin
  • D𝑦=4𝑎3𝑥𝑎sin
  • E𝑦=𝑎4𝑎𝑥cos

Q12:

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

  • A𝑦=𝑥+𝑥9+3lnln
  • B𝑦=𝑥+𝑥93lnln
  • C𝑦=13𝑥9𝑥+1345lnln
  • D𝑦=13𝑥9𝑥+1345lnln
  • E𝑦=𝑥+𝑥923lnln

Q13:

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

  • A(𝑦+1)(1𝑥)=𝑒
  • B(𝑦1)(1+𝑥)=𝑒
  • C(𝑦+1)(1𝑥)=𝑒
  • D(𝑦1)(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)?

  • A𝑓(𝑥)
  • B𝑓(𝑥)
  • C𝑓(𝑥)
  • D𝑓(𝑥)

Q15:

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

  • A𝑓(𝑥)=3𝑥+3𝑥+4𝑥+3
  • B𝑓(𝑥)=9𝑥+6𝑥+4𝑥+6
  • C𝑓(𝑥)=27𝑥48𝑥634
  • D𝑓(𝑥)=3𝑥2𝑥6

Q16:

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

  • A8𝑥+12cot
  • B8𝑥833+12cot
  • C8𝑥12+833cot
  • D8𝑥12+833cot
  • E8𝑥12cot

Q17:

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

  • A𝑓(𝑡)=8𝑡+2𝑡6+83tansec
  • B𝑓(𝑡)=8𝑡+2𝑡83+2tansec
  • C𝑓(𝑡)=4𝑡+𝑡6+83tansec
  • D𝑓(𝑡)=2𝑡+8𝑡83+2tansec
  • E𝑓(𝑡)=2𝑡+8𝑡6+83tansec

Q18:

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

  • A𝑓(𝜃)=3𝜃2𝜃+3sincos
  • B𝑓(𝜃)=2𝜃3𝜃1sincos
  • C𝑓(𝜃)=3𝜃2𝜃3𝜃2sincos
  • D𝑓(𝜃)=3𝜃2𝜃3𝜃+8sincos
  • E𝑓(𝜃)=𝜃2𝜃3𝜃+2sincos

Q19:

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

  • A𝑓(𝑥)=4𝑥+8𝑥4𝑥1sin
  • B𝑓(𝑥)=2𝑥+8𝑥+𝑥+1sin
  • C𝑓(𝑥)=4𝑥8𝑥4𝑥1sin
  • D𝑓(𝑥)=2𝑥+8𝑥4𝑥1sin
  • E𝑓(𝑥)=4𝑥8𝑥4𝑥+1sin

Q20:

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

  • A𝑦=355𝑥+922𝑥+10120sincos
  • B𝑦=92𝑥35𝑥+13sincos
  • C𝑦=925𝑥+352𝑥+8920sincos
  • D𝑦=352𝑥925𝑥+8920sincos
  • E𝑦=922𝑥355𝑥+8920sincos

Q21:

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

  • A𝑓(𝑥)=𝑥30+𝑥211𝑥573
  • B𝑓(𝑥)=𝑥30+𝑥2+11𝑥5+73
  • C𝑓(𝑥)=𝑥30+𝑥2+21𝑥5+13
  • D𝑓(𝑥)=𝑥5+16𝑥5
  • E𝑓(𝑥)=𝑥5+26𝑥5+13

Q22:

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

  • A𝑓(𝑥)=𝑥1+4344𝑥4434lnlnln
  • B𝑓(𝑥)=𝑥1+4344𝑥+434+4lnlnln
  • C𝑓(𝑥)=𝑥1+4344𝑥+434+4lnlnln
  • D𝑓(𝑥)=4𝑥344𝑥434lnlnln
  • E𝑓(𝑥)=𝑥1+4344𝑥4434lnlnln

Q23:

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

  • A𝑓(𝑥)=4𝑥5𝑥+3𝑥+5
  • B𝑓(𝑥)=4𝑥5𝑥+3𝑥+2
  • C𝑓(𝑥)=𝑥5𝑥3+3𝑥2+5𝑥+2
  • D𝑓(𝑥)=𝑥5𝑥3+3𝑥2+2𝑥+5
  • E𝑓(𝑥)=12𝑥10𝑥+3𝑥+2𝑥+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.

  • A𝑥8
  • B𝑥+12
  • C2𝑥
  • D𝑥

Q25:

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

  • A𝑠(𝑡)=99𝑡44sin
  • B𝑠(𝑡)=99𝑡44sin
  • C𝑠(𝑡)=49𝑡44sin
  • D𝑠(𝑡)=49𝑡44sin

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