Solving simple differential equations

Solving the Differential Equation y+y=0

Problem: Solve the differential equation y+y=0.

Explanation:

The equation is a second-order linear homogeneous differential equation with constant coefficients. Its general form is y+y=0.

First, we assume a solution of the form y(x)=erx.

Upon differentiating, we find:

  • First derivative: y(x)=rerx
  • Second derivative: y(x)=r2erx

Substituting into the original equation, we get (r2+1)erx=0.

Since erx is never zero, r2+1=0, which leads to r=±i where i is the imaginary unit.

Therefore, the general solution becomes y(x)=Acos(x)+Bsin(x).

About Linear Independence:

The functions cos(x) and sin(x) are linearly independent because there is no constant C such that cos(x)=C×sin(x) for all x.

Solving the Differential Equation y+5y+12=0

Problem: Solve the differential equation y+5y+12=0.

Explanation:

The equation is a second-order linear homogeneous differential equation with constant coefficients. Its general form is y+5y+12=0.

First, we assume a solution of the form y(x)=erx.

Upon differentiating, we find:

  • First derivative: y(x)=rerx
  • Second derivative: y(x)=r2erx

Substituting into the original equation, we get r2erx+5erx+12=0.

Since erx is never zero, we get r2+17=0, leading to r=±i17.

Therefore, the general solution becomes y(x)=Acos(17x)+Bsin(17x).