<p>In this paper we use the idea of normal family to find out the possible solution of the following special case of algebraic differential equation <Equation ID="Equ116"> <EquationSource Format="TEX">\( P_k\big (z,f,f^{(1)},\ldots , f^{(k)}\big )=f^{(1)}(f-\mathscr {L}_k(f))-\varphi (f-a)(f-b)=0, \)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>P</mi> <mi>k</mi> </msub> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">(</mo> </mrow> <mi>z</mi> <mo>,</mo> <mi>f</mi> <mo>,</mo> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo maxsize="1.2em" minsize="1.2em" stretchy="true">)</mo> </mrow> <mo>=</mo> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>-</mo> <msub> <mi mathvariant="script">L</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>-</mo> <mi>φ</mi> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>-</mo> <mi>a</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo>-</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mrow> </math></EquationSource> </Equation>where <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathscr {L}_k(f)= \sum _{i=0}^k a_i f^{(i)}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="script">L</mi> <mi>k</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msubsup> <mo>∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>0</mn> </mrow> <mi>k</mi> </msubsup> <msub> <mi>a</mi> <mi>i</mi> </msub> <msup> <mi>f</mi> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\varphi \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>φ</mi> </math></EquationSource> </InlineEquation> is an entire function, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(a_i\in \mathbb {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>i</mi> </msub> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\((i=0,1,\ldots , k)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>i</mi> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>k</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(a_k=1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>a</mi> <mi>k</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(a, b\in \mathbb {C}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo>∈</mo> <mi mathvariant="double-struck">C</mi> </mrow> </math></EquationSource> </InlineEquation> such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(a\ne b\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>a</mi> <mo>≠</mo> <mi>b</mi> </mrow> </math></EquationSource> </InlineEquation>. The obtained results improve and generalise the results of Li and Yang [<CitationRef CitationID="CR11">11</CitationRef>] and Xu et&#xa0;al. [<CitationRef CitationID="CR23">23</CitationRef>] in a large scale.</p>