Answer to Question #102843 in Differential Equations for Ajay

Question

d^2y/dx^2–cotxdy/dx–sinx^2 y=cosx–cos^3x

Answer 1

SOLVE d ² y/d x ²-cot x d y/d x-sin ² x y=cos x-cos ³ x

Solution

P=- cot x Q=- sin ² x R=cos x-cos ³ x

choose , z such that

(dz/d x) ² = sin ² x

taking square root on both sides we get

dz/d x=sin x

∫ dz/d x =∫ sin x d x

z= - cos x

z =- cos x

derivative with respect x

dz/d x= sin x

d ² z/d x ²= cos x

P ₁= (d ² z/d x ²+ P dz/d x )/(dz/d x) ²

P ₁=cos x-cot x sin x/sin ² x

P ₁= {cos x-(cos x/sin x)sin x}/sin ² x

P ₁=cos x-cos x/sin ² x

P ₁=0

Q ₁=Q/(dz/d x)²

Q ₁= - sin ² x/sin ² x

R ₁ = Q/(dz/d x) ²

R ₁ = (cos x-cos 3 x)/sin ² x

R ₁= -z-(-z ³)/{1-(-z)²}

R ₁= -z+z ³/1-z ²

R ₁= - z(-z ²+1)/(1-z ²)

R ₁ = -z

Reduced Equation is

d ² y/d z ²-y= -z .......>(1)

Complementary solution:-

Auxiliary Equation is

m ² -1=0

m=±1

y _c(x)=C ₁ e ^z+C ₂ e ^-z

Particular Solution:-

y _p(x)= -A z (2)

derivative w.r.t z

d y/dz= -A

again derivative

d ² y/d z ²=0

by putting the value in equation no 1

0+A z=-z

A=-1

putting in equation no 2

y _p(x)= z

y(x)=y _c(x)+y _p(x)

y(x)=C ₁ e ^z+C ₂ e ^-z +z

Hence The complete solution is

y(x)=C ₁ e ^z+ C ₂ e ^-z+z

y(x)=C ₁ e ^{-cos x}+ C ₂ e ^{cos x} -cos x Answer