Question

y'' - 3y' - 10y = 0 ; y(0)=1 and y'(0)=10

Answer 1

Corresponding (auxiliary) equation

$$r^2-3r-10=0$$

$$r_1=-2, r_2=5$$

The general solution of the homogeneous differential equation is

$$y=c_1e^{-2x}+c_2e^{5x}$$

$$y(0)=c_1+c_2=1$$

$$y'=-2c_1e^{-2x}+5c_2e^{5x}$$

$$y'(0)=-2c_1+5c_2=10$$

$$c_1=-\dfrac{5}{7}, c_2=\dfrac{12}{7}$$

$$y=-\dfrac{5}{7}e^{-2x}+\dfrac{12}{7}e^{5x}$$