Question

solve the differential equation :- d²y/dx² -4dy/dx+3y =x​

Answer 1

Step-by-step explanation:

Can anyone solve d²y/dx² + y = e^x?

This is a linear differential equation of order [math]2[/math]. The solution to such differential equations consists of [math]2[/math] parts, the complimentary function, [math]y_c[/math], and the particular integral, [math]y_p[/math].

[math]y_c[/math] is given by solving the D.E. [math]\frac {d^2 y}{dx^2} + y = 0[/math]. This has the general solution [math]c_1 cos (x) + c_2 sin (x)[/math]. (Same as SHM)

The particular integral is obtained by operating [math]\frac {1}{\frac {d^2}{dx^2} + 1}[/math] on [math]e^x[/math].

Fortunately, we have short-cut methods for few functions and [math]e^x[/math] is one of them. Using the short-cut method, we get the particular integral as[math] \frac {1}{1^2 + 1} e^x[/math] i.e. [math]\frac {e^x}{2}[/math].

Hence the general solution is