Differential equation

[pine terrace]

How can I solve this differential equation? Looks like a linear differential equation but the form of the homogeneous equation solutions is not given, so idk what to do

[frank helm]

hint: product rule

[pine terrace]

I don't really know what to do with this hint

[gloomy edge]

let u=dy/dx
(2+x⁴)du/dx - 4x³u = e^x(x⁴-4x³+2)

isolate your derivative:
and separate
du/dx - 4x³u/(x⁴+2) = e^x(x⁴-4x³+2)/(x⁴+2)
multiply by the integrating factor and apply product rule:
μ(du/dx - 4x³u/(x⁴+2)) = dμ/dx(u)+du/dx(μ)

now you isolate dμ/dx=-4x³μ/(x⁴+2)
and integrate:
μ=1/(x⁴+2)

∫dμu/dxdx=∫(μe^x(x⁴-4x³+2)/(x⁴+2)dx
μu=∫(μe^x(x⁴-4x³+2)/(x⁴+2)dx
solve for u and substitute μ=1/(x⁴+2)

u=(x⁴+2)∫(e^x(x⁴+4x³+2)/(x⁴+2)²)dx
and evaluate:
u=(x⁴+2)(e^x/(x⁴+2)+C1)
and expand:
u=x⁴C1+2C1+e^x
recall u=dy/dx
and integrate:
y=C1x⁵/5 + 2xC1 + e^x + C2

we know that if x=0, y=1
1=0+0+e⁰+C2
C2=0
and if x=1, y=11+e
11+e=C1/5 + 2C1 + e
C1/5 + 2C1 = 11
C1=5

therefore
y=x⁵+10x+e^x

@pine terrace

[pine terrace]

thank you