Question

Solve the initial value problem yy′+x = √x² + y² with y(2)= −√21

To solve this, we should use the substitution
u =
u' =

Answer 1

To solve the initial value problem involving a differential equation with the given condition y(2) = -√21, a substitution method is typically used, and the correct substitution must be determined for simplification.

Step-by-step explanation:

The initial value problem given is yy′ + x = √(x² + y²) with the condition y(2) = -√21. To solve such a differential equation, a common method involves substituting components of the equation with another variable to simplify the problem. The student hints at a substitution method but doesn't provide the necessary substitution expressions. An example of such a substitution could be letting u = y², after which taking the derivative concerning x gives u′ = 2yy′. This process could potentially simplify the original equation into a solvable form.

However, it is essential to determine the correct substitution based on the characteristics of the given equation. After applying the substitution, the resulting equation can be solved using standard techniques for ordinary differential equations and then reversing the substitution to find y(x).