2WBB0Calculus I

2WBB0 · Topic 08

First-order differential equations

Recognize separable and linear equations, solve systematically, and enforce initial conditions.

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What you need to be able to do

  • Classify equations as separable or first-order linear.
  • Use an integrating factor after normalizing the coefficient of y′.
  • Determine the integration constant and verify the result.
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1. Classifying and separating

Classification decides the method. A separable equation can be rearranged into a product of a y-only differential and an x-only differential.

Dividing by a y-expression can discard equilibrium solutions. Check values that make the divided factor zero before continuing.

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Exam method

  • Try to isolate y′ and factor the right side into x-only times y-only.
  • Record equilibrium solutions before dividing by h(y).
  • Move all y terms with dy and all x terms with dx.
  • Integrate both sides, solve for y, then apply the condition.
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Classifying and separating: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

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Classifying and separatingSelf-review

Worked example 1

Solve y′=xy.y'=xy.
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2. The integrating-factor algorithm

A first-order linear equation must first be written with coefficient 1 in front of y′. Only then is p(x) read off.

The integrating factor is chosen so the entire left side becomes one product derivative. This is the reason the method works, not an extra formula to memorize.

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Exam method

  • Divide by the coefficient of y′.
  • Identify p(x) and q(x).
  • Compute μ=e^(integral p) and multiply the whole equation by μ.
  • Recognize (μy)′, integrate, and divide by μ.
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The integrating-factor algorithm: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

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The integrating-factor algorithmSelf-review

Worked example 1

Solve for x>0x>0: xy′+2y=8x2−10.xy'+2y=8x^2-10.
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3. Initial conditions and verification

The general solution contains a constant. Substitute the initial point only after the general form is correct, unless a definite-integral form is deliberately used.

Verification is fast: differentiate the proposed solution, substitute y and y′ into the original equation, then test the initial condition separately.

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Exam method

  • Keep C visible through all simplification.
  • Substitute the initial x and y values and solve one scalar equation for C.
  • Differentiate the final expression.
  • Check both the differential equation and the initial condition.
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Initial conditions and verification: worked examples

2 questions

Attempt each problem before revealing the complete in-app solution.

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Initial conditions and verificationSelf-review

Worked example 1

For the solution y=2x2−5+C/x2y=2x^2-5+C/x^2 with y(1)=0y(1)=0, find C.
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Formulas used in this topic

  • y′+p(x)y=q(x),μ(x)=e∫p(x) dxy'+p(x)y=q(x),\qquad \mu(x)=e^{\int p(x)\,dx}
  • (μy)′=μq,y=∫μq dx+Cμ(\mu y)'=\mu q,\qquad y=\frac{\int\mu q\,dx+C}{\mu}
  • y′=g(x)h(y)⟹dyh(y)=g(x) dxy'=g(x)h(y)\Longrightarrow\frac{dy}{h(y)}=g(x)\,dx
  • y=yh+ypy=y_h+y_p
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