Solve the differential equation and find all the real solutions of the differential equation.
The solution is .
Consider the linear differential operator
The characteristic polynomial of is defined as
The characteristic polynomial of the operator as follows.
Then the characteristic polynomial is as follows.
Solve the characteristic polynomial and find the roots as follows.
Therefore, the roots of the characteristic polynomials are and .
Since, the roots of the characteristic equation is different real numbers.
The exponential functions and form a basis of the kernel of T .
Hence, they form a basis of the solution space of the homogenous differential equation is T (f) = 0 .
The complementary function as follows.
The particular solution to the differential equation is of the form role="math" localid="1660806105789" .
Differentiate the function with respect to t as follows.
Similarly, differentiate the function with respect to t as follows.
Substitute the values for x" and for x in x"+2x as follows.
Substitute the value for in x"+2x = cos(t) as follows.
Compare the coefficients of Cost and sint as follows.
A = 1
B = 0
Substitute the value 1 for A and 0 for B in role="math" localid="1660806778527" as follows.
Therefore, the general solution as follows.
Thus, the general solution of the differential equation is .
Consider a quadratic form of two variables. Consider the system of differential equations
Or more sufficiently
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