Maths4U Original · Learn, check, explain
Use an integrating factor and determine the constant from initial data.
For \(y'+P(x)y=Q(x)\), an integrating factor is \(I=e^{\int P(x)dx}\). Multiplication gives \((Iy)'=IQ\), allowing one integration followed by the initial condition.
Use one integration constant, and verify the result by substituting it into both the differential equation and the initial condition. Restrict the solution to an interval where the coefficients and chosen integrating factor are defined.
Solve \(y'+1y=5e^{4x},\ y(0)=6\). Write the solution as \(y=Ce^{-1x}+Ke^{4x}\).
\((1+4)K=5\implies K=1\)
Answer: \(1\).
\(C=6-K=5\)
Answer: \(5\).
\(y=e^{4 x} + 5 e^{- x}\); evaluate at \(\displaystyle x=1\).
Answer: \(56.4375472390015\).
\(y'(0)=5-1(6)=-1\)
Answer: \(-1\).
Solve \(y'+2y=6e^{1x},\ y(0)=7\). Write the solution as \(y=Ce^{-2x}+Ke^{1x}\).
\((2+1)K=6\implies K=2\)
Answer: \(2\).
\(C=7-K=5\)
Answer: \(5\).
\(y=2 e^{x} + 5 e^{- 2 x}\); evaluate at \(\displaystyle x=1\).
Answer: \(6.11324007310115\).
\(y'(0)=6-2(7)=-8\)
Answer: \(-8\).
Solve \(y'+3y=7e^{2x},\ y(0)=8\). Write the solution as \(y=Ce^{-3x}+Ke^{2x}\).
\(\displaystyle (3+2)K=7\implies K=\frac{7}{5}\)
Answer: \(\displaystyle \frac{7}{5}\).
\(\displaystyle C=8-K=\frac{33}{5}\)
Answer: \(\displaystyle \frac{33}{5}\).
\(\displaystyle y=\frac{7 e^{2 x}}{5} + \frac{33 e^{- 3 x}}{5}\); evaluate at \(\displaystyle x=1\).
Answer: \(10.6732731897308\).
\(y'(0)=7-3(8)=-17\)
Answer: \(-17\).
Write complete solutions before opening the worked solution. The marking guidance rewards reasoning as well as answers.
Use an integrating factor and determine the constant from initial data.
Try the problem & review your workingUse an integrating factor and determine the constant from initial data.
Try the problem & review your workingApply the method to recent source-labelled questions from this chapter.
Browse recent examination questions · All topic resources and the full bankNext skill: Second order differential equations: The homogeneous case →