Question

HELLO

I AM CURRENTLY IN DIFFERENTIAL EQUATIONS

PLEASE EXPLAIN EACH STEP SO I CAN LEARN FROM YOU (I KNOW SOME PEOPLE ONLY CARE ABOUT TEH ANSWER, BUT WILL REALLY APPRECIATE IT) TO SAVE TIME FEEL FREE TO JUST SAY A LAW, THEOREM, OR CONCEPT FOR AN EXPLANATION AND I CAN GO AHEAD AND STUDY IT ON MY OWN.

i REALLY DO READ THESE VERY CAREFULLY AND USE THE COMMENTS OFTEN, SO JUST A LITTLE HEADS UP.

I FIND IT DIFFICULT TO READ HANDWRITTEN STUFF SO IF YOU CAN DO IT ELECTRONICALLY THAT WOULD BE FANTASTIC.

PLEASE TAKE YOUR TIME AND STAY SAFE OUT THERE.

Use variation of parameters to find the general solution of the differential equation: 4y" = x2

Answer 1

Solution Griven, 44" = X² (1) Y" = x² 1 will colm First will solve the homogenous part of (1), that y"=0 (2) Dd dx Auxiliary 1949 tion or D²y=0 => m=0,O y = (a + (₂x) ex y = (1+C2x), where c, and ez are arbitary constants compairing (3) with y=c4+ (zu gives U=1, U=a Then general solution of (2) ip given by, y = cofitp. I (4) Pos-particular C.F. Complementary function, where Integral (of. = cru+qu and P. I = 4 f(x) + u g(x) where f(x) = -suR dx da and gra)=sur dx Also wie the wrong Rian of u and u lip. u du W (4,W) = V = a y ax da - W(1.) Voil

compairing (1) with, y" + Py't Qy = R gives 2 R = x² f(x) =-) wR de xixdx -sx.x² -S v=x,W(1,2)=1=W and Rad? 03 da *** f(x) 9-(1) = sar da -5.2.2 {: "=1, W=1, R=x2 8(2) from (4) general polution of (1) is y = Gute ut u f(x) tug(2) =(61+ (2x) +11 (24) y =((, tax) 4 x4 3 OR 3 74 4 3 (Citeza) + x4 34 (4-3) OR y = (Cot CqX) + x4 12 Asy