A Bernoulli differential equation is one of the formdydx+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−𝑛 transforms the Bernoulli equation into the linear equationdudx+(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 equationy′−7xy=y4x12,𝑦′−7𝑥𝑦=𝑦4𝑥12,and find the solution that satisfies y(1)=1.

The given differential equation is y′−7xy=y4x12. This is a Bernoulli equation where n=4, P(x)=-7x, and Q(x)=x12.

To solve this, we first perform a substitution with u=y1−n = y^-3. Then, du/dx = -(1-n)y^-n * dy/dx = 3y^-4 * dy/dx.

Substituting these into the original equation, we get:

3y^-4 * dy/dx + 7x * y^-3 = x12.

Substituting u=y^-3, we get:

(1/3) * du/dx + 7x * u = x12.

This is a linear differential equation which can be solved using an integrating factor. The integrating factor is e^(∫7xdx) = e^(7/2 * x^2).

Multiplying the entire equation by the integrating factor, we get:

e^(7/2 * x^2) * du/dx + 7x * e^(7/2 * x^2) * u = x12 * e^(7/2 * x^2).

The left side of this equation is the derivative of (u * e^(7/2 * x^2)) with respect to x. So, integrating both sides with respect to x, we get:

u * e^(7/2 * x^2) = ∫x12 * e^(7/2 * x^2) dx.

To solve the integral on the right, we can use the method of substitution with v=7/2 * x^2, dv=7x dx, and dx=dv/(7x). The integral becomes:

∫x11 * e^v * dv/7.

This integral can be solved using the method of integration by parts or tabular integration. After solving the integral, substitute back v=7/2 * x^2 and u=y^-3 to get the solution for y.

Finally, use the initial condition y(1)=1 to find the constant of integration.