If q=0, you can divide variables and integrate both parts.

dx/P(x)=dt/(p-bt).

Dear Tanmay,

using MAPLE I obtain the following solution where I denotes pure imaginary number.

To obtain simpler expression I think we must specify the time interval (is p-bt positive or negative ?)

x(t) = 1/2*((I*(-b^2+4*c*a)^(1/2)+b)*BesselJ(-I/b/(I*b-(-b^2+4*c*a)^(1/2))*(-4*c*a+b^2+I*b*(-b^2+4*c*a)^(1/2)),2*(1/b^2*(-p+b*t)*q*a)^(1/2))-(I*(-b^2+4*c*a)^(1/2)+b)*_C1*BesselY(-I/b/(I*b-(-b^2+4*c*a)^(1/2))*(-4*c*a+b^2+I*b*(-b^2+4*c*a)^(1/2)),2*(1/b^2*(-p+b*t)*q*a)^(1/2))-2*b*(1/b^2*(-p+b*t)*q*a)^(1/2)*(_C1*BesselY(4*I*a*c/b/(I*b-(-b^2+4*c*a)^(1/2)),2*(1/b^2*(-p+b*t)*q*a)^(1/2))-BesselJ(4*I*a*c/b/(I*b-(-b^2+4*c*a)^(1/2)),2*(1/b^2*(-p+b*t)*q*a)^(1/2))))/a/(_C1*BesselY(-I/b/(I*b-(-b^2+4*c*a)^(1/2))*(-4*c*a+b^2+I*b*(-b^2+4*c*a)^(1/2)),2*(1/b^2*(-p+b*t)*q*a)^(1/2))-BesselJ(-I/b/(I*b-(-b^2+4*c*a)^(1/2))*(-4*c*a+b^2+I*b*(-b^2+4*c*a)^(1/2)),2*(1/b^2*(-p+b*t)*q*a)^(1/2)))

Dear sir,

p-bt is +ve and i forgot to mention p>=bt>=0  

 The solution by Mathematica 9 is attached,

Best Luck