x²d²y/dx² + 3xdy/dx +y = 1/(1-x²)
Answers
Answer:This equation can be solved by Euler Cauchy's Method:
Substitute x=e^t in the above equation;
On simplifying, the above equation reduces to:
(D²-D-3D+4)y=2e^(3t); where D= d/dt and is known as differential operator.
The general solution of the above equation consists of complementary function and the Particular integral.
The complementary function(CF) is the solution of the homogeneous equation:
(D²-4D+4)y=0 => D²-4D+4=0
D=2,2
Hence Complementary function is y=(A+Bt)e^2t
where A and B are arbitrary constants.
The particular integral (PI) is given by:
y= 2e^(3t)/(D²-4D+4);
To find the PI, put D=3 in above equation:
y= 2e^(3t)/1 =>y= 2e^(3t);
Hence the general solution is
y=C.F. + P.I.
y=(A+Bt)e^2t+ 2e^(3t)
Resubstitute x=e^t;
y= x^(2)(A+Bt)+(x^3)/2
Step-by-step explanation:
Answer:
x²d²y/dx² + 3xdy/dx +y = 1/(1-x²)