91Ó°ÊÓ

An implicit solution is one where the relationship between the dependent variable, say \( y \), and the independent variable, \( x \), is not solved explicitly for \( y \). Instead, \( y \) is embedded in an equation along with \( x \). Implicit solutions can be especially useful when it is difficult or impossible to solve for \( y \) directly. For the given differential equation, an implicit solution is derived after integrating both sides of the equation:

• Left side integrates to: \( \frac{1}{2} \arctan(2y) + C_1 \)
• Right side integrates to: \(-\frac{1}{2} \ln(1 + x^4) + C_2 \)

The implicit form of the solution combines these integrated results: \( \arctan(2y) = -\ln(1 + x^4) + C \). This equation relates \( y \) and \( x \) indirectly, hence remaining implicit.
To make this form satisfy the initial condition \( y(1) = 0 \), solve for \( C \) as \( \ln(2) \), leading to the complete implicit solution: \[ \arctan(2y) = -\ln(1 + x^4) + \ln(2) \].

An explicit solution expresses the dependent variable \( y \) directly in terms of the independent variable \( x \). Unlike an implicit solution, explicit solutions provide a more straightforward relationship between the variables. For the differential equation given, the explicit solution is achieved by solving the implicit form for \( y \). Starting from: \( \arctan(2y) = \ln\left(\frac{2}{1 + x^4}\right) \),we first take the tangent of both sides: \[ 2y = \tan \left( \ln \left( \frac{2}{1 + x^4} \right) \right) \].Finally, dividing by 2 gives the explicit solution: \[ y = \frac{1}{2} \tan \left( \ln \left( \frac{2}{1 + x^4} \right) \right) \].This results in a clear \( y = f(x) \) form, illustrating how \( y \) changes with \( x \) explicitly.

Separation of variables is a fundamental technique for solving differential equations, particularly useful when dealing with simple ordinary differential equations. It involves rearranging an equation so that each variable, along with its differential, is on a separate side of the equation. In the problem provided, the given differential equation is:\[(1 + x^4)dy + x(1 + 4y^2)dx = 0\].By rearranging terms, variables are separated: \[ \frac{dy}{1 + 4y^2} = -\frac{x}{1 + x^4} dx \].This separation allows us to integrate both sides independently. The process is crucial because it transforms a differential equation into a form that is readily integrable, making it easier to solve whether implicitly or explicitly for particular solutions.

An initial-value problem (IVP) in differential equations specifies not only the equation but also an initial condition that the solution must satisfy. This initial condition is essential for determining the specific solution among a family of possible solutions. In the exercise provided, the IVP is given with:\( y(1) = 0 \).This condition requires that whatever solutions we determine must work at \( x = 1 \) to produce \( y = 0 \). When solved for in either implicit or explicit form, applying the initial condition yields the particular constant \( C \) that ensures the solution matches the initial value. For our implicit solution: \[ \arctan(2y) = -\ln(1 + x^4) + C \],using the initial condition \( y(1) = 0 \) allows us to find \( C = \ln(2) \), ensuring the solution meets the required initial value constraints.