Solve 1st Order ODE Homework - Get Help Now

[manimaran1605]

Homework Statement

Solve the below differential equation

Homework Equations

The Attempt at a Solution

I have attached my attempt at solution. But I don't how to get rid of (ln y) term in my equation i.e, i Don't know how to write in terms of y. Please help

 

[slider142]

It is not true that $$\int \frac{d^2 y}{y} = \ln y + C.$$ What is true is that $$\int \frac{dy}{y} = \ln y + C.$$ Since we do not have the form $$\frac{dy}{y}$$ anywhere in our equation, we cannot apply that integral to this equation.
The standard method with which we solve this type of differential equation (second order linear homogeneous) is to assume the solution is a linear combination of exponential functions of the form y_k = e^kx where k may be a complex number, and substitute this assumption into the equation in order to solve for the various possible values of k.
That is, if you find $y_1 = e^{k_1x}$ and $y_2 = e^{k_2x}$ both satisfy the differential equation, then $y = C_1e^{k_1x} + C_2e^{k_2x}$ also satisfies the original equation for any particular pair of values $C_1$ and $C_2$.
However, another plausible method is that you may already know two functions whose second derivative yields the negation of the original function. It then stands to reason that any linear combination of those two functions solves this equation.