Ficha 3- Exercicios sobre Equacoes diferenciais homogeneas e exactas

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2° tipo : Equações Homogêneas 
Resolva as seguintes equações : 
1) (𝑥2 − 𝑦2)𝑑𝑥 − 2𝑥𝑦𝑑𝑦 = 0 
2) (2𝑥 − 𝑦)𝑑𝑥 − (𝑥 + 4𝑦)𝑑𝑦 = 0 
3) (𝑥2 − 𝑦2)𝑑𝑥 − 𝑥𝑦𝑑𝑦 = 0 
4) (𝑥2 − 3𝑦2)𝑑𝑥 − 𝑥𝑦𝑑𝑦 = 0, 𝑐𝑜𝑚 𝑦 = 1 𝑒 𝑥 = 2 
5) (𝑥 − 𝑦)𝑑𝑥 + 𝑥𝑦𝑑𝑦 = 0 
6) 𝑥𝑑𝑥 + (𝑦 − 2𝑥)𝑑𝑦 = 0 
7) (𝑦2 − 𝑦𝑥)𝑑𝑥 − 𝑥2𝑑𝑦 = 0 
8) 
𝑑𝑦
𝑑𝑥
=
𝑦−𝑥
𝑦+𝑥
 
9) −𝑦𝑑𝑥 + (𝑥 + √𝑥𝑦)𝑑𝑦 = 0 
10) 2𝑥2𝑦𝑑𝑥 = (3𝑥3 + 𝑦3)𝑑𝑦 = 0 
11) 
𝑑𝑦
𝑑𝑥
=
𝑦
𝑥
+
𝑥
𝑦
 
12) 𝑦
𝑑𝑥
𝑑𝑦
= 𝑥 + 4𝑦𝑒
−2𝑥
𝑦 
13) (𝑦 + 𝑥𝑐𝑜𝑡𝑔
𝑦
𝑥
) 𝑑𝑥 − 𝑥𝑑𝑦 = 0 
14) (𝑥2 + 𝑥𝑦 − 𝑦2)𝑑𝑥 + 𝑥𝑦𝑑𝑦 = 0 
15) 𝑥𝑦2
𝑑𝑦
𝑑𝑥
= 𝑦3 − 𝑥3 , 𝑦(1) = 2 
16) 2𝑥2
𝑑𝑦
𝑑𝑥
= 3𝑥𝑦 − 𝑦2 , 𝑦(1) = −2 
17) (𝑥 + 𝑦𝑒
𝑦
𝑥) 𝑑𝑥 − 𝑥𝑒
𝑦
𝑥 𝑑𝑦 = 2 , 𝑦(1) = 0 
18) (𝑦2 + 3𝑥𝑦)𝑑𝑥 = (4𝑥2 + 𝑥𝑦)𝑑𝑦 , 𝑦(1) = 1 
19) (𝑥 + √𝑥𝑦)
𝑑𝑦
𝑑𝑥
+ 𝑥 − 𝑦 = 𝑥−
1
2 𝑦
3
2 , 𝑦(1) = 1 
20) 𝑦2𝑑𝑥 + (𝑥2 + 𝑥𝑦 + 𝑦2)𝑑𝑦 = 0 , 𝑦(0) = 1 
21) (𝑥 + √𝑦2 − 𝑥𝑦)
𝑑𝑦
𝑑𝑥
= 𝑦 , 𝑦 (
1
2
) = 1 
 
4° tipo : Equações Exactas 
Resolva as seguintes equações diferenciais: 
1) (𝑥2 − 𝑦2)𝑑𝑥 − 2𝑥𝑦𝑑𝑦 = 0 
2) (2𝑥 − 𝑦 + 1)𝑑𝑥 − (𝑥 + 3𝑦 − 2)𝑑𝑦 = 0 
3) 𝑒𝑦𝑑𝑥 + (𝑥𝑒𝑦 − 2𝑦)𝑑𝑦 = 0 
4) (𝑥3 + 𝑦2)𝑑𝑥 + (2𝑥𝑦 + 𝑐𝑜𝑠𝑦)𝑑𝑦 = 0 
5) [𝑦𝑐𝑜𝑠(𝑥𝑦) +
𝑦
√𝑥
] 𝑑𝑥 + [𝑥𝑐𝑜𝑠(𝑥𝑦) + 2√𝑥 +
1
𝑦
] 𝑑𝑦 = 0 
6) (2𝑥 − 1)𝑑𝑥 + (3𝑦 + 7)𝑑𝑦 = 0 
7) (5𝑥 + 4𝑦)𝑑𝑥 + (4𝑥 − 8𝑦3)𝑑𝑦 = 0 
8) (2𝑦2𝑥 − 3)𝑑𝑥 + (2𝑦𝑥2 + 4)𝑑𝑦 = 0 
9) (3𝑥2𝑦 − 4𝑙𝑛𝑥)𝑑𝑥 + (𝑥3 − 𝑙𝑛𝑦)𝑑𝑦 = 0 
10) (𝑦3 + 𝑦2𝑠𝑒𝑛𝑥 − 𝑥)𝑑𝑥 + (3𝑥𝑦2 + 2𝑦𝑐𝑜𝑠𝑥)𝑑𝑦 = 0 
11) 
𝑑𝑦
𝑑𝑥
=
2+𝑦𝑒𝑥𝑦
2𝑦−𝑥𝑒𝑥𝑦
 
12) (4𝑥3𝑦 − 15𝑥2 − 𝑦)𝑑𝑥 + (𝑥4 + 3𝑦2 − 𝑥)𝑑𝑦 = 0 
13) (𝑥 + 𝑦)2𝑑𝑥 + (2𝑥𝑦 + 𝑥2 − 1)𝑑𝑦 = 0 , 𝑦(1) = 1 
14) (4𝑦 + 2𝑥 − 5)𝑑𝑥 + (6𝑦 + 4𝑥 − 1)𝑑𝑦 = 0 , 𝑦(−1) = 2 
15) (1 −
3
𝑥
+ 𝑦) 𝑑𝑥 + (1 −
3
𝑦
+ 𝑥) 𝑑𝑦 = 0 
16) (𝑥2𝑦3 −
1
1+9𝑥2
)
𝑑𝑥
𝑑𝑦
+ 𝑥3𝑦2 = 0 𝑦𝑐𝑜𝑠(𝑥𝑦) +
𝑦
√𝑥
 
17) (𝑡𝑔𝑥 − 𝑠𝑒𝑛𝑥𝑠𝑒𝑛𝑦)𝑑𝑥 + 𝑐𝑜𝑠𝑥𝑐𝑜𝑠𝑦𝑑𝑦 = 0 
18) (1 − 2𝑥2 − 2𝑦)
𝑑𝑦
𝑑𝑥
= 4𝑥3 + 4𝑥𝑦 
19) (𝑦2𝑐𝑜𝑠𝑥 − 3𝑥2𝑦 − 2𝑥)𝑑𝑥 + (2𝑦𝑠𝑒𝑛𝑥 − 𝑥3 + 𝑙𝑛𝑦)𝑑𝑦 = 0 
20) 𝑥
𝑑𝑦
𝑑𝑥
= 2𝑥𝑒𝑥 − 𝑦 + 6𝑥2