Solve (3x + 2) ^ 2 * (d ^ 2 * y)/(d * x ^ 2) + 5(3x + 2) * derivative of y for x - 3y = x ^ 2 + x + 1

Question

\((3x + 2)^2\frac{d^2y}{dx^2}+5(3x+2)\frac{dy}{dx}\) derivative of y for x - 3y = x² + x + 1.

Answer 1

 \((3x + 2)^2\frac{d^2y}{dx^2}+5(3x+2)\frac{dy}{dx}-3y=x^2+x+1\) 

3x + 2 = t

\(\frac{dy}{dx}=\frac{dy}{dt}.\frac{dt}{dx}=3\frac{dy}{dt}\)

\(∴\frac{d^2y}{dx^2}=\frac{d}{dx}(\frac{dy}{dx})=\frac{d}{dt}(\frac{3dy}{dt})\frac{dt}{dx}=9\frac{d^2y}{dt^2}\)

\(∴9t^2\frac{d^2y}{dt^2}+15+\frac{dy}{dt}-3y=\frac{1}{9}(t^2-4t+4)+\frac{t-2}{3}+1\)

\(⇒9t^2\frac{d^2y}{dt^2}+15t\frac{dy}{dt}-3y=\frac{1}{9}(t^2-t+7)\)

Let t = e^z

\(t\frac{d}{df}=\frac{d}{dz}=D\)

\(t^2\frac{d^2}{df^2}=D(D-1)\)

∴ (9D (D-1) + 15 D - 3) y = 1/9 (t² - t + 7)

⇒ (9D2 + 6D - 3)y = 1/9(t² - t + 7)

It's auxilarly equation : 

9m² + 6m - 3 = 0

⇒ (m+1) (9m-3) = 0

⇒ (m+1) (3m-1) = 0

⇒ m = -1 , 1/3

∴ C.F = c₁e^-z + c₂e^z/3 = c₁/t + c₂t^1/3

\(P.I=\frac{1}{9D^2+6D-3}\,\frac{1}{9}\,(t^2-t+7)\)

\(=\frac{1}{-27}(1-(9D^2+6D))^{-1}(t^2-t+7)\)

\(=\frac{-1}{27}(1+(9D^2+6D)+(9D^2+6D)^2+...)(t^2-t+7)\)

\(=\frac{-1}{27}(t^2-t+7+18+12t-6+72)\)

\(=\frac{-1}{27}(t^2+11t+91)\)

y = C.F + P.I

\(=\frac{C_1}{t}+C_2t^{\frac{1}{3}}-\frac{1}{27}(t^2+11t+91)\)

\(=\frac{C_1}{3x+2}+C_2(3x+2)^{1/3}-\frac{1}{27}((3x+2)^2+11(3x+2)+91)\)

\(=\frac{C_1}{3x+2}+C_2(3x+2)^{1/3}-\frac{1}{27}(9x^2+45x+117)\)

which is complete solution of given differential equation.