The portal has been deactivated. Please contact your portal admin.

Lesson Worksheet: Linear Homogeneous Differential Equations with Constant Coefficients Mathematics

In this worksheet, we will practice solving second- and higher-order linear homogeneous differential equations with constant coefficients.

Q1:

Suppose there are differentiable functions 𝐶 and 𝑆, defined for all real numbers and satisfying 𝐶=𝑆 and 𝑆=𝐶.

From the derivatives, what can you deduce about 𝐶𝑆?

  • A𝐶𝑆 is a constant function.
  • B𝐶𝑆 is an exponential function.
  • C𝐶𝑆 is a hyperbolic function.
  • D𝐶𝑆 always equals 0.
  • E𝐶𝑆 is a trigonometric function.

It is known that every function that satisfies 𝑦=𝑦 is a combination 𝑎𝑒+𝑏𝑒 for some constants 𝑎 and 𝑏. Find all functions 𝐶 satisfying 𝐶(0)=5 and 𝐶𝑆=24, where 𝑆=𝐶.

  • A𝐶(𝑥)=2𝑒3𝑒 and 𝐶(𝑥)=3𝑒+2𝑒
  • B𝐶(𝑥)=2𝑒3𝑒 and 𝐶(𝑥)=3𝑒2𝑒
  • C𝐶(𝑥)=2𝑒+3𝑒 and 𝐶(𝑥)=3𝑒2𝑒
  • D𝐶(𝑥)=2𝑒+3𝑒 and 𝐶(𝑥)=3𝑒+2𝑒
  • E𝐶(𝑥)=5𝑒+5𝑒 and 𝐶(𝑥)=5𝑒5𝑒

Sketching graphs of the functions above suggest that 𝐶(𝑥)=𝐴(𝑥𝑘)cosh for some constants 𝐴 and 𝑘. Find these constants, where 𝑘>0.

  • A𝐴=26, 𝑘=32ln
  • B𝐴=26, 𝑘=322lnln
  • C𝐴=5, 𝑘=32ln
  • D𝐴=5, 𝑘=22ln
  • E𝐴=5, 𝑘=322lnln

Q2:

The functions 𝑦=𝑒, 𝑦=𝑒, and 𝑦=𝑒 are three linearly independent solutions of the differential equation 𝑦+2𝑦𝑦2𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=1, 𝑦(0)=2, and 𝑦(0)=0.

  • A𝑦=43𝑒13𝑒
  • B𝑦=43𝑒+13𝑒
  • C𝑦=43𝑒13𝑒
  • D𝑦=𝑒+𝑒𝑒
  • E𝑦=43𝑒+13𝑒

Q3:

The functions 𝑦=𝑒, 𝑦=𝑒, and 𝑦=𝑒 are three linearly independent solutions of the differential equation 𝑦6𝑦+11𝑦6𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=0, 𝑦(0)=0, and 𝑦(0)=3.

  • A𝑦=32𝑒3𝑒+32𝑒
  • B𝑦=34𝑒+32𝑒34𝑒
  • C𝑦=12𝑒𝑒+12𝑒
  • D𝑦=3𝑒+3𝑒
  • E𝑦=𝑒+𝑒

Q4:

The functions 𝑦=𝑒, 𝑦=𝑥𝑒, and 𝑦=𝑥𝑒 are three linearly independent solutions of the differential equation 𝑦3𝑦+3𝑦𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=2, 𝑦(0)=0, and 𝑦(0)=0.

  • A𝑦=2𝑒+2𝑥𝑒+𝑥𝑒
  • B𝑦=2𝑒2𝑥𝑒+𝑥𝑒
  • C𝑦=2𝑒2𝑥𝑒𝑥𝑒
  • D𝑦=2𝑒+2𝑥𝑒𝑥𝑒
  • E𝑦=2𝑒+2𝑥𝑒𝑥𝑒

Q5:

The functions 𝑦=𝑒, 𝑦=𝑒, and 𝑦=𝑥𝑒 are three linearly independent solutions of the differential equation 𝑦5𝑦+8𝑦4𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=1, 𝑦(0)=4, and 𝑦(0)=0.

  • A𝑦=12𝑒+13𝑒10𝑥𝑒
  • B𝑦=12𝑒13𝑒+10𝑥𝑒
  • C𝑦=12𝑒+13𝑒10𝑥𝑒
  • D𝑦=12𝑒13𝑒10𝑥𝑒
  • E𝑦=12𝑒+13𝑒+10𝑥𝑒

Q6:

The functions 𝑦=1, 𝑦=3𝑥cos, and 𝑦=3𝑥sin are three linearly independent solutions of the differential equation 𝑦+9𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=3, 𝑦(0)=1, and 𝑦(0)=2.

  • A𝑦=299293𝑥133𝑥cossin
  • B𝑦=299+293𝑥133𝑥cossin
  • C𝑦=299+293𝑥+133𝑥cossin
  • D𝑦=299133𝑥293𝑥cossin
  • E𝑦=299+133𝑥293𝑥cossin

Q7:

The functions 𝑦=𝑒, 𝑦=𝑒𝑥cos, and 𝑦=𝑒𝑥sin are three linearly independent solutions of the differential equation 𝑦3𝑦+4𝑦2𝑦=0(). Find a particular solution satisfying the initial conditions 𝑦(0)=1, 𝑦(0)=0, and 𝑦(0)=0.

  • A𝑦=2𝑒𝑒𝑥𝑒𝑥cossin
  • B𝑦=2𝑒+𝑒𝑥𝑒𝑥cossin
  • C𝑦=𝑒𝑥+𝑒𝑥cossin
  • D𝑦=2𝑒𝑒𝑥𝑒𝑥cossin
  • E𝑦=2𝑒+𝑒𝑥+𝑒𝑥cossin

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