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Solve the differential equation dy/dx = x^2 + y^2.

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What’s inside this PDF?

Question: Solve the differential equation dy/dx = x^2 + y^2.

Options:

  1. y = x^3/3 + C
  2. y = x^2 + C
  3. y = x^2 + x + C
  4. y = Cx^2 + C

Correct Answer: y = x^3/3 + C

Solution: 

This is a non-linear differential equation. The solution can be found using substitution methods.

Solve the differential equation dy/dx = x^2 + y^2.

Practice Questions

Q1
Solve the differential equation dy/dx = x^2 + y^2.
  1. y = x^3/3 + C
  2. y = x^2 + C
  3. y = x^2 + x + C
  4. y = Cx^2 + C

Questions & Step-by-Step Solutions

Solve the differential equation dy/dx = x^2 + y^2.
  • Step 1: Identify the type of differential equation. Here, dy/dx = x^2 + y^2 is a non-linear equation.
  • Step 2: Consider a substitution method. A common substitution for equations like this is to let y = vx, where v is a function of x.
  • Step 3: Differentiate y = vx with respect to x to find dy/dx. This gives dy/dx = v + x(dv/dx).
  • Step 4: Substitute dy/dx into the original equation: v + x(dv/dx) = x^2 + (vx)^2.
  • Step 5: Simplify the equation. This will lead to a new equation in terms of v and x.
  • Step 6: Solve the new equation for v. This may involve separating variables or using another method.
  • Step 7: Once you find v, substitute back to find y in terms of x.
  • Non-linear Differential Equations – The equation involves both the dependent variable y and its derivative in a non-linear manner.
  • Substitution Methods – Techniques used to simplify the equation for easier solving, often involving a change of variables.
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