Which of the following five differential equations cannot be solved using an integrating factor? (More than one correct answer.)

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To determine whether an integrating factor can be used, we first recall that the standard integrating factor technique applies to first-order linear ODEs of the form y' + p(x) y = q(x).

Option a: y' + sqrt(y) = x^2. This is not linear in y because of the sqrt(y) term. The integrating factor method hinges on linearity in y, so this equation cannot be solved by the integrating factor approach in its usual form. In fact, it is a nonlinear first-order equation, which generally requires other methods (e.g., substitution, numerical methods, or qualitative analysis) rather than an integrating factor.

Option b: y'' + y = sin(x). This is a linear second-order ODE with constant coefficients. However, the integrating factor method is specifically designed for first-order linear equations. While we can solve this equation using standard methods (characteristic equation for the homogeneous part and a particular solution for the nonhomogeneous part), the typical integrating factor technique does not apply directly to second-order equations in the same straightforward way. Thus, using an integrating factor in the standard sense is not the appropriate approach here.

Option c: y' + 2y = cos(x). This is a textbook first-order linear ODE with p(x) = 2. An integrating factor μ(x) = e^{∫2 dx} = e^{2x} leads to (e^{2x} y)' = e^{2x} cos(x), which can be integrated to solve for y. Therefore, this equation can be solved by the integrating factor method.

Option d: y' + y/x = ln(x). This is also a first-order linear ODE, with p(x) = 1/x. The integrating factor is μ(x) = e^{∫(1/x) dx} = e^{\ln x} = x. Multiplying through by x gives (x y)' = x ln(x), which integrates to solve for y. Hence, this equation is solvable via integrating factor.

Option e: y' + x y = x^2. This fits the first-order linear form y' + p(x) y = q(x) with p(x) = x and q(x) = x^2. The integrating factor is μ(x) = e^{∫ x dx} = e^{x^2/2}. Multiplying through yields (e^{x^2/2} y)' = e^{x^2/2} x^2, which can be integrated to find y. Thus, integrating factor applies here as well.

In summary, the integrating factor technique is applicable to options c, d, and e because they are first-order linear ODEs. It is not applicable in the standard way to option a due to nonlinearity in y, and it is not the typical method for option b because that is a second-order equation, for which the integrating factor is not the standard tool. The phrasing of the question (more than one correct answer) aligns with identifying the equations not solvable by the integrating factor approach in its conventional form, which are options a and b.

MTH1030 -1035 - S1 2025 MTH1030/35 Week 11 lesson quiz: Differential equations

Which of the following five differential equations cannot be solved using an integrating factor? (More than one correct answer.)

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