Solve the differential equation (y')^7 - (y')^5 + y' + 3 = 0

Solve the differential equation $(y')^7 - (y')^5 + y' + 3 = 0$.

Explanation of Method

The equation involves only $y'$ with no explicit $x$ or $y$ dependence. Setting $p=y'$ reduces the problem to finding the real roots of a degree-7 polynomial $p^7-p^5+p+3=0$; each real root $p_i$ gives a family of straight-line solutions $y=p_ix+C$.

Set up the polynomial in $p=y'$

Let $p = y' = \dfrac{dy}{dx}$. The equation becomes

\[p^7 - p^5 + p + 3 = 0.\]

This is a seventh-degree polynomial in $p$, so by the Abel–Ruffini theorem no general closed-form (radical) expression for its roots exists.

Rule out rational roots

By the Rational Root Theorem, any rational root must divide $3$, i.e. lie in $\{\pm1,\pm3\}$. Direct substitution gives $f(-1)=2$, $f(1)=4$, $f(-3)=-1944$, $f(3)=1950$, none zero. So the polynomial has no rational roots.

Locate the real roots

Let $f(p)=p^7-p^5+p+3$. A sign check gives $f(-2)=-95<0$ and $f(0)=3>0$, so by the Intermediate Value Theorem there is a real root between $-2$ and $0$. Applying Sturm's theorem (or a numerical root solver such as sympy.Poly.nroots) to the full degree-7 polynomial shows this is the only real root; the other six roots form three complex-conjugate pairs. High-precision root finding gives

\[p_0 \approx -1.2521412990911069,\]

which satisfies $f(p_0)=0$ to the precision shown. (Note: a coarse Newton-Raphson estimate can easily land on a nearby but slightly inaccurate value such as $-1.2207$; the value above was verified numerically to 20 digits.)

Integrate each real root

For any root $p_i$ of $f(p)=0$ (real or complex), $y'=p_i$ is a constant, so integrating directly gives $y = p_i x + C_i$. Since the ODE has no $x,y$ dependence, this constant-slope family is always a valid solution branch whenever $f(p)=0$ has a real solution.

Result

Final answer:

The general solution consists of straight lines $y = p_i x + C$, one family for each real root $p_i$ of $p^7-p^5+p+3=0$. This polynomial has exactly one real root (confirmed by Sturm's theorem), so there is exactly one real solution family:

\[\boxed{y = p_0\,x + C, \qquad p_0 \approx -1.2521412990911069 \ \text{(the unique real root of } p^7-p^5+p+3=0\text{)}}\]

The remaining six roots of the polynomial are complex (three conjugate pairs) and do not yield additional real solution families.