Question

Writing Prompt(s)

One method for solving a system of first order linear differential equation such as

x ′ = a x + b y y ′ = c x + d y

is to take the derivative of the first equation and use the second equation to ``decouple'' the system and create a second order equation, which we can solve using our previous techniques. Does this always work? If not, what conditions on the constants a, b, c, and d must be enforced? If it does work, we can then arrive at an equation for x(t). How do we proceed in finding an equation for y(t)?

Answer 1

Using the method of elimination,the system of n linear differential equations reduces into a single nth order linear differential equation.This method is useful for simple cases like system of order 2.
Now the given system is
x'=ax+by
y'=cx+dy
Where a, b, c, d are constant coefficients.
In the method of elimination there is no such restriction on constant coefficient a,b,c,d.But if you wish to find some non trivial(non-constant solution) solution to the system one of them must be non-zero.
In method of elimination we do similar things as we do to find solution to system of equation in linear algebra.
In this method we first differentiate first equation and substitute y' from the second equation and we get a second order linear differential in x which is
x"-(a+d)x'+(ad-bc)=0
Now this equation could be solve easily by finding the roots of the characteristic equation
m²-(a+d)m+(ad-bc)=0
Let us suppose the general solution to x be
x=C₁f₁+C₂f₂ where C₁ and C₂ are constants f₁, f₂ are parts of the solution and the function of independent variable only.
Then by putting value of x in y'=cx+dy we get
y'=c(C₁f₁+C₂f₂)+dy
=>y'-dy=c(C₁f₁+Cf₂)
Which is a first order first degree linear differential equation and finding the integrating factor I.F. and multiplying the equation with this I.F. ,then integrating we get the required solution to y.(solving for y using integrating factor is one method to solve, you could find other methods to as for your comfortable).