Find the general solution to the differential equation; y'' + 4y' = 24x^2

The first step in tackling any second order differential equation is to find the Auxiliary Function. Looking at the equation's left hand side, we can find the derivative terms - we will substitute the derivative terms with another variable m, which we can then solve to find a suitable Complementary Function (CF). For second order terms, or any y'' terms, we use m², then use m for first order terms (y'). If zero order terms exist, then only the coefficient will remain. Hence, y'' + 4y' becomes m² + 4m. We then attempt to solve this auxiliary function by solving for m² + 4m = 0, for which we get m = 0, -4. In this case, since we have two unique terms (we can also get a repeated solution and complex conjugate solutions), we can say that the CF is equal to Ae^(m1)x + Be^(m2)x, where A and B are constants. In this problem, our complementary function is A + Be^-4x.
Since this is a non-homogeneous equation, we must consider the non-derivative x term on the right hand side of the equation - we need to form a Particular Integral (PI) that will add to the CF to give the final solution. Since there is no x² term in the CF, we can assign the form of the PI as px³ + qx² + rx. This particular integral will now need to substituted back into the differential equation to find the value of the constants p, q and r. The first and second derivatives of the PI are 3px² + 2qx + r and 6px + 2q respectively. Substituting this into the differential equation gives 6px + 2q + 4(3px² + 2qx + r) = 24x². By equating the coefficients of each term, we find that p = 2, q = -1.5, r = 0.75. This gives our final PI as 2x³ - 1.5x² + 0.75x.
The final step in this question is to add together the CF and PI to give the general solution. Therefore, the general solution is: Y = A + Be^-4x + 2x³ - 1.5x² + 0.75x. The reason this is the general solution and not a particular solution is because it still has the constants A and B - you require at least two sets of limits to solve for those constants.