help with nonlinear differential equation?

To verify that a function is a solution to a differential equation, all you have to do is differentiate the function an appropriate number of times (depending on the order of the DE), plug those expressions into the DE, and see if the equation is satisfied.

For the first purported solution we have that:

y(x) = exp(x) = y'(x) = y''(x)

Plugging y(x) and y''(x) into the differential equation, we get:

(exp(x))^2 ?=? (exp(x))^2

which is obviously correct, so this is a solution.

Similarly, for the second solution:

y(x) = cos(x)

y'(x) = -sin(x)

y''(x) = -cos(x)

Plugging into the DE:

(-cos(x))^2 ?=? (cos(x))^2

This is also correct, so this is also a solution.

Now, let's try a linear combination of the two solutions. For a linear DE, a linear combination of solutions is also a solution, but this is a nonlinear DE, so let's see.

y(x) = a*exp(x) + b*cos(x)

y'(x) = a*exp(x) - b*sin(x)

y''(x) = a*exp(x) - b*cos(x)

(a*exp(x) - b*cos(x))^2 ?=? (a*exp(x) + b*cos(x))^2

(a^2)*exp(2x) - a*b*exp(x)*cos(x) + (b^2)*(cos(x))^2 ?=? (a^2)*exp(2x) -+a*b*exp(x)*cos(x) + (b^2)*(cos(x))^2

- a*b*exp(x)*cos(x) ?=? a*b*exp(x)*cos(x)

In general this is not true, so unlike the case for a linear DE, a linear combination of the solutions to a nonlinear DE is not necessarily a solution.

Use an integrating ingredient to locate the equation for the final answer: dy / dt = a million / (t + y) dt / dy = t + y dt / dy - t = y dt / dy + P(y)t = f(y) P(y) = -a million f(y) = y I(y) = e ^ ? P(y) dy I(y) = e ^ ? -a million dy I(y) = e ^ (-y) I(y)t = ? I(y)f(y) dy te ^ (-y) = ? ye ^ (-y) dy combine this function on the main marvelous area by ability of things: ? ye ^ (-y) dy enable f'(y) = e ^ (-y) f(y) = -e ^ (-y) enable g(y) = y g'(y) = a million ? f'(y)g(y) dy = f(y)g(y) - ? f(y)g'(y) dy ? ye ^ (-y) dy = -ye ^ (-y) + ? e ^ (-y) dy ? ye ^ (-y) dy = -ye ^ (-y) - e ^ (-y) + C ? ye ^ (-y) dy = -(y + a million)e ^ (-y) + C ? ye ^ (-y) dy = -(y + a million) / e ^ y + C Now plug this quintessential into the equation to get the final answer: te ^ (-y) = ? ye ^ (-y) dy te ^ (-y) = -(y + a million) / e ^ y + C t = -(y + a million) + Ce ^ y t = Ce ^ y - y - a million finally remedy for the consistent to locate the particular answer: whilst t = -a million, y = 0 -a million = C - a million C = 0 t = -y - a million y = -t - a million