Question

1. Identify order, degree, linear or nonlinearities of the following differential equations:

a) ( √[4]left(d⁴ y/dx⁴right)⁵ + 5x d³ y/dx³ + y = √[3]left(d³ y/dx³right)³ + y² )
b) ( (y''+y')³ = x (xy''+y)⁻⁴ )

2. Find the differential equation whose solution is ((c+x) y³=x⁴³).

3. Solve the following differential equation by using an appropriate method:

a) ( 1/x³ dy/dx + 1+y²/y⁴ (1+x²) = 0 )
b) ( (2+y²)dx = (xy + 2y + y³)dy )

Answer 1

The first differential equation is nonlinear of degree 5, while the second equation is a nonlinear equation of unknown degree. To find the differential equation whose solution is ((c+x) y³=x⁴³), differentiate both sides and simplify. To solve the differential equation (1/x³ dy/dx + 1+y²/y⁴ (1+x²) = 0), separate the variables and integrate both sides. The equation (2+y²)dx = (xy + 2y + y³)dy is a nonlinear equation of degree 3.

Step-by-step explanation:

The first differential equation, a), can be identified as a nonlinear differential equation of degree 5. The equation contains square roots and powers of y, which make it nonlinear. The second differential equation, b), is also a nonlinear equation of unknown degree because it contains an inverse power term.

To find the differential equation whose solution is ((c+x) y³=x⁴³), we need to differentiate both sides of the equation with respect to x. After differentiating and simplifying, we get (3(c+x)²y² + (c+x)³(3y² dy/dx) = 437x³).

To solve the differential equation (1/x³ dy/dx + 1+y²/y⁴ (1+x²) = 0), we rewrite it as (dy/dx + x/(x³+y²) + y²/(y⁴ (1+x²)) = 0). This equation is separable, so we can separate the variables and integrate both sides to find the solution.

The equation (2+y²)dx = (xy + 2y + y³)dy is a nonlinear differential equation of degree 3. It is not linear because the variables x and y appear in higher powers and are multiplied together.