Differential equations

Question

A Bernoulli differential equation is one of the form 𝑑𝑦𝑑𝑥+𝑃(𝑥)𝑦=𝑄(𝑥)𝑦𝑛.dydx+P(x)y=Q(x)yn.

Observe that, if 𝑛=0n=0 or 11, the Bernoulli equation is linear. For other values of 𝑛n, the substitution 𝑢=𝑦1−𝑛u=y1−ntransforms the Bernoulli equation into the linear equation 𝑑𝑢𝑑𝑥+(1−𝑛)𝑃(𝑥)𝑢=(1−𝑛)𝑄(𝑥).dudx+(1−n)P(x)u=(1−n)Q(x).

Use an appropriate substitution to solve the equation 𝑦′−6𝑥𝑦=𝑦5𝑥11,y′−6xy=y5x11,

and find the solution that satisfies 𝑦(1)=1.y(1)=1. 

𝑦(𝑥)=y(x)= 

Answer 1

y' - 6xy = y⁵x; u = y^-4; u' = - 4y^-5y'; y' = - u'y⁵/4; - u'y⁵/4 - 6xy = y⁵x; - u'/4 - 6xu = x; u' + 24xu = - 4x

Integrated factor M = e^∫24xdx = e^{12x^2}; (u' + 24xu)e^{12x^2} = (-4x)e^{12x^2}; d/dx(ue^{12x^2}) = -4xe^{12x^2};

ue^{12x^2} = - 1/3e^{12x^2} + C; u = - 1/3 + Ce^{-12x^2} = y^-4; -1/3 + Ce^-12 = 1; C = 4/3e¹²