First-order equations and growth models

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Growth models are differential equations: the rate of change of a population depends on the population itself. This lesson solves first-order linear equations with an integrating factor, uses a given substitution to turn a non-linear equation into a linear one, and then reads long-term behaviour from a phase line without solving: equilibria, their stability, the carrying capacity of a logistic model and the effect of harvesting.

The integrating factor

A first-order linear equation has the form

dy/(dx) + p(x) y = q(x).

Multiply both sides by the integrating factor I(x) = e^(∫ p(x) dx). Since I' = pI, the left side becomes (d/(dx))(Iy) by the product rule, so

I(x) y = ∫ I(x) q(x) dx.

The constant of integration belongs on the right-hand side before you divide by I(x).

Solve dy/(dx) + (2/x)y = 3x for x > 0, given that y = 2 when x = 1.

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Find the integrating factor
Method
p(x) = 2/x, so I = e^(2 ln x) = x².
Reason
With x > 0, no modulus is needed in ln x.
Working
Answer
Multiplying through: x²dy/(dx) + 2xy = 3x³, that is (d/(dx))(x²y) = 3x³.
Integrate, then fit the condition
Method
x²y = (3/4)x⁴ + C.
Reason
Keep C before dividing, so that it is divided by x² too.
Working
Answer
At x = 1: 2 = 3/4 + C, so C = 5/4 and
y = 3x²/4 + 5/(4x²).

Quick check

Check: y' = (3/2)x-5/(2x³) and (2/x)y = (3/2)x + 5/(2x³); their sum is 3x. Writing y = (3/4)x² + C (adding C after dividing) would fail this check. The general solution is a family of curves, one for each C; the condition picks one.

Solve dy/(dx)-y/x = x² for x > 0, given that y = 1 when x = 1, and find y when x = 2.

y when x = 2

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Here p(x) = -1/x, so I = e^(- ln x) = 1/x and

(d/(dx))(y/x) = x,; y/x = x²/2 + C.

At x = 1, 1 = 1/2 + C, so C = 1/2 and y = (1/2)x³ + (1/2)x. At x = 2, y = 4 + 1 = 5.

A given substitution can make an equation linear

Some equations become linear after a substitution that the question supplies. Differentiate the substitution, replace y and dy/(dx) everywhere, and simplify to a standard form. Remember to return to the original variable. The same idea reduces some second-order equations to the constant-coefficient form.

Use the substitution z = 1/y to find the general solution of dy/(dx) = y-y² for y > 0.

The equation in z, its solution, and y

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dz/(dx) = -(1/y²)dy/(dx) = -(y-y²)/y² = 1-z, so dz/(dx) + z = 1. With integrating factor e^x: ze^x = e^x + C, so z = 1 + Ce^(-x) and

y = 1/(1 + Ce^(-x)).

Every solution with y > 0 tends to 1 as x → ∞.

Phase lines show long-term behaviour

For an equation dP/(dt) = f(P), the equilibria are the values where f(P) = 0. A phase line marks them on a number line and shows, between them, whether P increases (f > 0) or decreases (f < 0). An equilibrium is stable if nearby solutions move towards it, and unstable if they move away. A slope field shows the same information as short line segments with gradient f(P) at many points; it is a sketching aid only.

Exponential growth: dP/(dt) = kP gives P = P₀e^kt, with no limit for k > 0.
Logistic growth: dP/(dt) = rP(1-P/K) has equilibria 0 (unstable) and K (stable), the carrying capacity. For r = 1/2, K = 100 and P₀ = 20, the solution is P = 100/(1 + 4e^(-t/2)).

Three curves start at P = 20, 60 and 140 when t = 0. The first rises slowly, then steeply around t = 2.8, where it passes 50, then levels off near 100. The second rises and levels off near 100 sooner. The third falls from 140 towards 100. A horizontal reference line marks P = 100. — Solutions of dP/dt = ½P(1 − P/100) from three starting values. Populations below 100 rise towards it along an S-shaped curve; a population above 100 falls towards it. None crosses the equilibrium P = 100.

Figure data: Logistic solutions approach the carrying capacity
Series	t	P
P(0) = 20	0.00	20.0000
P(0) = 20	0.25	22.0751
P(0) = 20	0.50	24.3001
P(0) = 20	0.75	26.6727
P(0) = 20	1.00	29.1875
P(0) = 20	1.25	31.8365
P(0) = 20	1.50	34.6085
P(0) = 20	1.75	37.4890
P(0) = 20	2.00	40.4610
P(0) = 20	2.25	43.5046
P(0) = 20	2.50	46.5979
P(0) = 20	2.75	49.7176
P(0) = 20	3.00	52.8396
P(0) = 20	3.25	55.9395
P(0) = 20	3.50	58.9937
P(0) = 20	3.75	61.9801
P(0) = 20	4.00	64.8786
P(0) = 20	4.25	67.6713
P(0) = 20	4.50	70.3434
P(0) = 20	4.75	72.8832
P(0) = 20	5.00	75.2819
P(0) = 20	5.25	77.5339
P(0) = 20	5.50	79.6361
P(0) = 20	5.75	81.5884
P(0) = 20	6.00	83.3925
P(0) = 20	6.25	85.0523
P(0) = 20	6.50	86.5728
P(0) = 20	6.75	87.9606
P(0) = 20	7.00	89.2228
P(0) = 20	7.25	90.3672
P(0) = 20	7.50	91.4017
P(0) = 20	7.75	92.3346
P(0) = 20	8.00	93.1738
P(0) = 20	8.25	93.9272
P(0) = 20	8.50	94.6023
P(0) = 20	8.75	95.2061
P(0) = 20	9.00	95.7455
P(0) = 20	9.25	96.2265
P(0) = 20	9.50	96.6551
P(0) = 20	9.75	97.0365
P(0) = 20	10.00	97.3756
P(0) = 20	10.25	97.6768
P(0) = 20	10.50	97.9441
P(0) = 20	10.75	98.1813
P(0) = 20	11.00	98.3916
P(0) = 20	11.25	98.5779
P(0) = 20	11.50	98.7429
P(0) = 20	11.75	98.8890
P(0) = 20	12.00	99.0182
P(0) = 20	12.25	99.1326
P(0) = 20	12.50	99.2337
P(0) = 20	12.75	99.3232
P(0) = 20	13.00	99.4022
P(0) = 20	13.25	99.4721
P(0) = 20	13.50	99.5338
P(0) = 20	13.75	99.5884
P(0) = 20	14.00	99.6366
P(0) = 20	14.25	99.6791
P(0) = 20	14.50	99.7167
P(0) = 20	14.75	99.7499
P(0) = 20	15.00	99.7793
P(0) = 20	15.25	99.8051
P(0) = 20	15.50	99.8280
P(0) = 20	15.75	99.8482
P(0) = 20	16.00	99.8660
P(0) = 60	0.00	60.0000
P(0) = 60	0.25	62.9592
P(0) = 60	0.50	65.8241
P(0) = 60	0.75	68.5780
P(0) = 60	1.00	71.2071
P(0) = 60	1.25	73.7006
P(0) = 60	1.50	76.0508
P(0) = 60	1.75	78.2529
P(0) = 60	2.00	80.3050
P(0) = 60	2.25	82.2074
P(0) = 60	2.50	83.9628
P(0) = 60	2.75	85.5754
P(0) = 60	3.00	87.0509
P(0) = 60	3.25	88.3959
P(0) = 60	3.50	89.6178
P(0) = 60	3.75	90.7246
P(0) = 60	4.00	91.7243
P(0) = 60	4.25	92.6250
P(0) = 60	4.50	93.4347
P(0) = 60	4.75	94.1611
P(0) = 60	5.00	94.8116
P(0) = 60	5.25	95.3932
P(0) = 60	5.50	95.9124
P(0) = 60	5.75	96.3753
P(0) = 60	6.00	96.7875
P(0) = 60	6.25	97.1542
P(0) = 60	6.50	97.4802
P(0) = 60	6.75	97.7697
P(0) = 60	7.00	98.0266
P(0) = 60	7.25	98.2544
P(0) = 60	7.50	98.4564
P(0) = 60	7.75	98.6353
P(0) = 60	8.00	98.7937
P(0) = 60	8.25	98.9339
P(0) = 60	8.50	99.0580
P(0) = 60	8.75	99.1678
P(0) = 60	9.00	99.2648
P(0) = 60	9.25	99.3507
P(0) = 60	9.50	99.4265
P(0) = 60	9.75	99.4936
P(0) = 60	10.00	99.5528
P(0) = 60	10.25	99.6052
P(0) = 60	10.50	99.6514
P(0) = 60	10.75	99.6922
P(0) = 60	11.00	99.7283
P(0) = 60	11.25	99.7601
P(0) = 60	11.50	99.7883
P(0) = 60	11.75	99.8131
P(0) = 60	12.00	99.8350
P(0) = 60	12.25	99.8544
P(0) = 60	12.50	99.8715
P(0) = 60	12.75	99.8866
P(0) = 60	13.00	99.8999
P(0) = 60	13.25	99.9116
P(0) = 60	13.50	99.9220
P(0) = 60	13.75	99.9312
P(0) = 60	14.00	99.9392
P(0) = 60	14.25	99.9464
P(0) = 60	14.50	99.9527
P(0) = 60	14.75	99.9582
P(0) = 60	15.00	99.9631
P(0) = 60	15.25	99.9675
P(0) = 60	15.50	99.9713
P(0) = 60	15.75	99.9747
P(0) = 60	16.00	99.9776
P(0) = 140	0.00	140.0000
P(0) = 140	0.25	133.7152
P(0) = 140	0.50	128.6198
P(0) = 140	0.75	124.4351
P(0) = 140	1.00	120.9621
P(0) = 140	1.25	118.0543
P(0) = 140	1.50	115.6018
P(0) = 140	1.75	113.5207
P(0) = 140	2.00	111.7454
P(0) = 140	2.25	110.2242
P(0) = 140	2.50	108.9157
P(0) = 140	2.75	107.7865
P(0) = 140	3.00	106.8092
P(0) = 140	3.25	105.9614
P(0) = 140	3.50	105.2244
P(0) = 140	3.75	104.5823
P(0) = 140	4.00	104.0223
P(0) = 140	4.25	103.5329
P(0) = 140	4.50	103.1049
P(0) = 140	4.75	102.7301
P(0) = 140	5.00	102.4016
P(0) = 140	5.25	102.1134
P(0) = 140	5.50	101.8605
P(0) = 140	5.75	101.6383
P(0) = 140	6.00	101.4430
P(0) = 140	6.25	101.2713
P(0) = 140	6.50	101.1202
P(0) = 140	6.75	100.9873
P(0) = 140	7.00	100.8703
P(0) = 140	7.25	100.7672
P(0) = 140	7.50	100.6765
P(0) = 140	7.75	100.5965
P(0) = 140	8.00	100.5261
P(0) = 140	8.25	100.4640
P(0) = 140	8.50	100.4092
P(0) = 140	8.75	100.3610
P(0) = 140	9.00	100.3184
P(0) = 140	9.25	100.2809
P(0) = 140	9.50	100.2478
P(0) = 140	9.75	100.2186
P(0) = 140	10.00	100.1929
P(0) = 140	10.25	100.1702
P(0) = 140	10.50	100.1502
P(0) = 140	10.75	100.1325
P(0) = 140	11.00	100.1169
P(0) = 140	11.25	100.1032
P(0) = 140	11.50	100.0910
P(0) = 140	11.75	100.0803
P(0) = 140	12.00	100.0709
P(0) = 140	12.25	100.0625
P(0) = 140	12.50	100.0552
P(0) = 140	12.75	100.0487
P(0) = 140	13.00	100.0430
P(0) = 140	13.25	100.0379
P(0) = 140	13.50	100.0335
P(0) = 140	13.75	100.0295
P(0) = 140	14.00	100.0261
P(0) = 140	14.25	100.0230
P(0) = 140	14.50	100.0203
P(0) = 140	14.75	100.0179
P(0) = 140	15.00	100.0158
P(0) = 140	15.25	100.0139
P(0) = 140	15.50	100.0123
P(0) = 140	15.75	100.0109
P(0) = 140	16.00	100.0096
Carrying capacity P = 100	0.00	100.0000
Carrying capacity P = 100	16.00	100.0000

Figure data: Logistic solutions approach the carrying capacity
Series	t	P
P(0) = 20	0.00	20.0000
P(0) = 20	0.25	22.0751
P(0) = 20	0.50	24.3001
P(0) = 20	0.75	26.6727
P(0) = 20	1.00	29.1875
P(0) = 20	1.25	31.8365
P(0) = 20	1.50	34.6085
P(0) = 20	1.75	37.4890
P(0) = 20	2.00	40.4610
P(0) = 20	2.25	43.5046
P(0) = 20	2.50	46.5979
P(0) = 20	2.75	49.7176
P(0) = 20	3.00	52.8396
P(0) = 20	3.25	55.9395
P(0) = 20	3.50	58.9937
P(0) = 20	3.75	61.9801
P(0) = 20	4.00	64.8786
P(0) = 20	4.25	67.6713
P(0) = 20	4.50	70.3434
P(0) = 20	4.75	72.8832
P(0) = 20	5.00	75.2819
P(0) = 20	5.25	77.5339
P(0) = 20	5.50	79.6361
P(0) = 20	5.75	81.5884
P(0) = 20	6.00	83.3925
P(0) = 20	6.25	85.0523
P(0) = 20	6.50	86.5728
P(0) = 20	6.75	87.9606
P(0) = 20	7.00	89.2228
P(0) = 20	7.25	90.3672
P(0) = 20	7.50	91.4017
P(0) = 20	7.75	92.3346
P(0) = 20	8.00	93.1738
P(0) = 20	8.25	93.9272
P(0) = 20	8.50	94.6023
P(0) = 20	8.75	95.2061
P(0) = 20	9.00	95.7455
P(0) = 20	9.25	96.2265
P(0) = 20	9.50	96.6551
P(0) = 20	9.75	97.0365
P(0) = 20	10.00	97.3756
P(0) = 20	10.25	97.6768
P(0) = 20	10.50	97.9441
P(0) = 20	10.75	98.1813
P(0) = 20	11.00	98.3916
P(0) = 20	11.25	98.5779
P(0) = 20	11.50	98.7429
P(0) = 20	11.75	98.8890
P(0) = 20	12.00	99.0182
P(0) = 20	12.25	99.1326
P(0) = 20	12.50	99.2337
P(0) = 20	12.75	99.3232
P(0) = 20	13.00	99.4022
P(0) = 20	13.25	99.4721
P(0) = 20	13.50	99.5338
P(0) = 20	13.75	99.5884
P(0) = 20	14.00	99.6366
P(0) = 20	14.25	99.6791
P(0) = 20	14.50	99.7167
P(0) = 20	14.75	99.7499
P(0) = 20	15.00	99.7793
P(0) = 20	15.25	99.8051
P(0) = 20	15.50	99.8280
P(0) = 20	15.75	99.8482
P(0) = 20	16.00	99.8660
P(0) = 60	0.00	60.0000
P(0) = 60	0.25	62.9592
P(0) = 60	0.50	65.8241
P(0) = 60	0.75	68.5780
P(0) = 60	1.00	71.2071
P(0) = 60	1.25	73.7006
P(0) = 60	1.50	76.0508
P(0) = 60	1.75	78.2529
P(0) = 60	2.00	80.3050
P(0) = 60	2.25	82.2074
P(0) = 60	2.50	83.9628
P(0) = 60	2.75	85.5754
P(0) = 60	3.00	87.0509
P(0) = 60	3.25	88.3959
P(0) = 60	3.50	89.6178
P(0) = 60	3.75	90.7246
P(0) = 60	4.00	91.7243
P(0) = 60	4.25	92.6250
P(0) = 60	4.50	93.4347
P(0) = 60	4.75	94.1611
P(0) = 60	5.00	94.8116
P(0) = 60	5.25	95.3932
P(0) = 60	5.50	95.9124
P(0) = 60	5.75	96.3753
P(0) = 60	6.00	96.7875
P(0) = 60	6.25	97.1542
P(0) = 60	6.50	97.4802
P(0) = 60	6.75	97.7697
P(0) = 60	7.00	98.0266
P(0) = 60	7.25	98.2544
P(0) = 60	7.50	98.4564
P(0) = 60	7.75	98.6353
P(0) = 60	8.00	98.7937
P(0) = 60	8.25	98.9339
P(0) = 60	8.50	99.0580
P(0) = 60	8.75	99.1678
P(0) = 60	9.00	99.2648
P(0) = 60	9.25	99.3507
P(0) = 60	9.50	99.4265
P(0) = 60	9.75	99.4936
P(0) = 60	10.00	99.5528
P(0) = 60	10.25	99.6052
P(0) = 60	10.50	99.6514
P(0) = 60	10.75	99.6922
P(0) = 60	11.00	99.7283
P(0) = 60	11.25	99.7601
P(0) = 60	11.50	99.7883
P(0) = 60	11.75	99.8131
P(0) = 60	12.00	99.8350
P(0) = 60	12.25	99.8544
P(0) = 60	12.50	99.8715
P(0) = 60	12.75	99.8866
P(0) = 60	13.00	99.8999
P(0) = 60	13.25	99.9116
P(0) = 60	13.50	99.9220
P(0) = 60	13.75	99.9312
P(0) = 60	14.00	99.9392
P(0) = 60	14.25	99.9464
P(0) = 60	14.50	99.9527
P(0) = 60	14.75	99.9582
P(0) = 60	15.00	99.9631
P(0) = 60	15.25	99.9675
P(0) = 60	15.50	99.9713
P(0) = 60	15.75	99.9747
P(0) = 60	16.00	99.9776
P(0) = 140	0.00	140.0000
P(0) = 140	0.25	133.7152
P(0) = 140	0.50	128.6198
P(0) = 140	0.75	124.4351
P(0) = 140	1.00	120.9621
P(0) = 140	1.25	118.0543
P(0) = 140	1.50	115.6018
P(0) = 140	1.75	113.5207
P(0) = 140	2.00	111.7454
P(0) = 140	2.25	110.2242
P(0) = 140	2.50	108.9157
P(0) = 140	2.75	107.7865
P(0) = 140	3.00	106.8092
P(0) = 140	3.25	105.9614
P(0) = 140	3.50	105.2244
P(0) = 140	3.75	104.5823
P(0) = 140	4.00	104.0223
P(0) = 140	4.25	103.5329
P(0) = 140	4.50	103.1049
P(0) = 140	4.75	102.7301
P(0) = 140	5.00	102.4016
P(0) = 140	5.25	102.1134
P(0) = 140	5.50	101.8605
P(0) = 140	5.75	101.6383
P(0) = 140	6.00	101.4430
P(0) = 140	6.25	101.2713
P(0) = 140	6.50	101.1202
P(0) = 140	6.75	100.9873
P(0) = 140	7.00	100.8703
P(0) = 140	7.25	100.7672
P(0) = 140	7.50	100.6765
P(0) = 140	7.75	100.5965
P(0) = 140	8.00	100.5261
P(0) = 140	8.25	100.4640
P(0) = 140	8.50	100.4092
P(0) = 140	8.75	100.3610
P(0) = 140	9.00	100.3184
P(0) = 140	9.25	100.2809
P(0) = 140	9.50	100.2478
P(0) = 140	9.75	100.2186
P(0) = 140	10.00	100.1929
P(0) = 140	10.25	100.1702
P(0) = 140	10.50	100.1502
P(0) = 140	10.75	100.1325
P(0) = 140	11.00	100.1169
P(0) = 140	11.25	100.1032
P(0) = 140	11.50	100.0910
P(0) = 140	11.75	100.0803
P(0) = 140	12.00	100.0709
P(0) = 140	12.25	100.0625
P(0) = 140	12.50	100.0552
P(0) = 140	12.75	100.0487
P(0) = 140	13.00	100.0430
P(0) = 140	13.25	100.0379
P(0) = 140	13.50	100.0335
P(0) = 140	13.75	100.0295
P(0) = 140	14.00	100.0261
P(0) = 140	14.25	100.0230
P(0) = 140	14.50	100.0203
P(0) = 140	14.75	100.0179
P(0) = 140	15.00	100.0158
P(0) = 140	15.25	100.0139
P(0) = 140	15.50	100.0123
P(0) = 140	15.75	100.0109
P(0) = 140	16.00	100.0096
Carrying capacity P = 100	0.00	100.0000
Carrying capacity P = 100	16.00	100.0000

Common mistakes

Using e^(∫ q(x) dx), or forgetting the integral, when forming the integrating factor e^(∫ p(x) dx).
Applying the integrating factor before writing the equation in the form dy/(dx) + p(x)y = q(x).
Adding the constant after dividing by the integrating factor.
Differentiating a substitution such as z = 1/y without the chain rule: dz/(dx) = -(1/y²)dy/(dx).
Calling an equilibrium stable without checking the sign of dP/(dt) on both sides of it.

Before you move on

A fish population (in thousands) is modelled by

dP/(dt) = (1/2)P(1-P/100)-12,

where 12 thousand fish are harvested per year. Find the equilibria and their stability, and describe what happens if P(0) = 30 and if P(0) = 50. What is the largest constant harvest that still allows an equilibrium?

Write your answer, then compare it with the solution

Equilibria, stability, both outcomes and the largest harvest

Hint 1

Clear the fractions

Multiplying f(P) = 0 by -200 gives P²-100P + 2400 = 0.

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Equilibria and stability
Method
P²-100P + 2400 = (P-40)(P-60) = 0.
Reason
f is a downward parabola in P, so it is positive only between its roots.
Working
Answer
f < 0 for P < 40, f > 0 for 40 < P < 60 and f < 0 for P > 60. So 40 is unstable and 60 is stable.
Read the two starting values
Method
Follow the arrows of the phase line.
Reason
Solutions cannot cross an equilibrium.
Working
Answer
From 30, P decreases and the population collapses: the model reaches 0 in finite time. From 50, P increases towards 60 thousand.
The largest sustainable harvest
Method
An equilibrium needs (1/2)P(1-P/100) = h to have a root.
Reason
The left side is greatest at P = 50, where it equals 12.5.
Working
Answer
So the largest constant harvest is 12.5 thousand fish per year. Any larger harvest makes dP/(dt) < 0 for every P.

Quick check

Harvesting at 12 turns the unharvested unstable equilibrium 0 into an unstable threshold at 40: a population below it is lost. The threshold is the practical lesson: a harvest that is safe for a large population can drive a smaller one to extinction.

Syllabus and review details

GCE A-Level H2 Further Mathematics 2027 · 2027

Last reviewed: 2026-11-07

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Guided steps

Reading settings

The integrating factor

A first-order linear equation

Find the integrating factor

Integrate, then fit the condition

Divide only after integrating

A given substitution can make an equation linear

A logistic-type equation made linear

Phase lines show long-term behaviour

Common mistakes

Before you move on

Harvesting a fish population

Equilibria and stability

Read the two starting values

The largest sustainable harvest

Lesson done

The integrating factor

A first-order linear equation

Divide only after integrating

A given substitution can make an equation linear

A logistic-type equation made linear

Phase lines show long-term behaviour

Common mistakes

Before you move on

Harvesting a fish population

Questions